diff b1f9bc4..56bc728 arithmeticity_local_field_lattices_v2.txt
diff --git a/arithmeticity_local_field_lattices_v2.txt b/arithmeticity_local_field_lattices_v2.txt
index a2453b7..ec76ea8 100644
--- a/arithmeticity_local_field_lattices_v2.txt
+++ b/arithmeticity_local_field_lattices_v2.txt
@@ -52,7 +52,7 @@ title[Arithmeticity of twisted finite correspondences]{Arithmeticity of twisted
Abstract:
We prove arithmetic exhaustion for bifinite correspondences between
arbitrarily scalar-twisted group factors of ICC groups commensurable with
-property-$(T)$ lattices over characteristic-zero local fields and twisted
+property-(T) lattices over characteristic-zero local fields and twisted
group factors of arbitrary countable ICC groups. Every such correspondence
is a closed summand of a finite direct sum of models obtained from actual
finite-index subgroup isomorphisms and finite-dimensional projective
@@ -67,7 +67,7 @@ algebraic relations, together with their unavoidable phases.
maketitle
tableofcontents
-bigskip
+
# A Guide for the Curious Reader
addcontentsline{toc}{section}{A Guide for the Curious Reader}
@@ -93,9 +93,9 @@ transformations leave the object unchanged.
Some of the most important groups are continuous: instead of
finitely many or countably many symmetries, they form a smooth
space. For example, the set of all rotations of the plane about the
-origin is a circle; the set of all invertible $ntimes n$ real matrices
-of determinant one, written $SL_n(mathbb R)$,bbR)$, is a continuous group of
-transformations of $mathbb R^n$bbR^n$ that preserve volume and orientation.
+origin is a circle; the set of all invertible ntimes n real matrices
+of determinant one, written SL_n(mathbb R),bbR), is a continuous group of
+transformations of mathbb R^nbbR^n that preserve volume and orientation.
Groups like these, that are also smooth manifolds, are called Lie
groups. They are the basic objects of the geometry of symmetry.
@@ -117,45 +117,45 @@ and they connect geometry, number theory, and group theory.
The groups in this paper are built by taking products of continuous
groups over different "places." The word place is from number
-theory: the real numbers $mathbb R$,bbR$, the complex numbers $mathbb C$,bbC$,
-and the $p$-adic numbers $mathbb Q_p$bbQ_p$ are all "completions" of the
+theory: the real numbers mathbb R,bbR, the complex numbers mathbb C,bbC,
+and the p-adic numbers mathbb Q_pbbQ_p are all "completions" of the
rationals, and a formula that holds over the rationals has a face at
each place. A "local field" here means a finite extension of
-$mathbb R$,bbR$, $mathbb C$,bbC$, or $mathbb Q_p$.bbQ_p$. Real place, complex place,
-$p$-adic place: these are the three kinds of ground fields on which our
+mathbb R,bbR, mathbb C,bbC, or mathbb Q_p.bbQ_p. Real place, complex place,
+p-adic place: these are the three kinds of ground fields on which our
groups are built. A big lattice can have several faces at once: a real
-face, and a $p$-adic face, and so on. The theory has to handle all of
+face, and a p-adic face, and so on. The theory has to handle all of
them, and one of the achievements here is that it does, uniformly.
-## 2. Rigidity: property $(T)$
+## 2. Rigidity: property (T)
-A group has Kazhdan's property $(T)$ if it is rigid: it cannot be
+A group has Kazhdan's property (T) if it is rigid: it cannot be
"almost" represented by symmetries without actually being represented.
Formally, every unitary representation that has almost-invariant unit
vectors already has an exactly invariant one. Intuitively, a group with
-property $(T)$ has a spectral gap: any attempt to wiggle a
+property (T) has a spectral gap: any attempt to wiggle a
representation produces a genuine, non-negligible displacement.
Rigidity is powerful. Many strong structural theorems in operator
-algebras and geometry hold only for groups with property $(T)$ (often,
+algebras and geometry hold only for groups with property (T) (often,
conjecturally, for all higher-rank lattices, an expectation this paper
-does not need). Property $(T)$ is the main structural hypothesis that
+does not need). Property (T) is the main structural hypothesis that
the "source" side of our theorem carries. It is what lets us pin down
the geometry of the correspondence rather than chase a wild object that
squirms away.
The class of groups that this paper can handle is written
-$mathscr K$,scrK$, and it consists of the countably infinite groups that are
+mathscr K,scrK, and it consists of the countably infinite groups that are
abstractly commensurable with a lattice in a product
-$G=prod_{i=1}^r G_i$, where each factor $G_i$ is the group of local
+G=prod_{i=1}^r G_i, where each factor G_i is the group of local
points of a connected, adjoint, absolutely simple algebraic group over a
-local field $k_iin{mathbb R,bbR,mathbb C,finitebbC,finite extension of mathbb Q_pbbQ_p}$,
+local field k_iin{mathbb R,bbR,mathbb C,finitebbC,finite extension of mathbb Q_pbbQ_p},
generated by the unipotent radicals of the proper parabolic subgroups,
-noncompact, and with property $(T)$. Abstractly commensurable means: an
+noncompact, and with property (T). Abstractly commensurable means: an
actual isomorphism between finite-index subgroups. No irreducibility,
cocompactness, product decomposition, or torsion hypothesis is imposed
on the lattice. The class is large; it contains the usual arithmetic
-lattices in products of higher-rank real and $p$-adic groups, the
+lattices in products of higher-rank real and p-adic groups, the
quaternionic and Cayley rank-one lattices, reducible lattices, and
nonuniform ones.
@@ -166,28 +166,28 @@ standard and holds for essentially all groups anyone cares about here.
## 3. Quantum shadows: group von Neumann algebras
-To every group $Gamma$ one can attach an algebra of operators that
+To every group Gamma one can attach an algebra of operators that
serves as the group's "quantum shadow." The construction goes as
-follows. Let $ell^2(Gamma)$ be the Hilbert space of square-summable
-functions on $Gamma$. For each $ginGamma$ there is a unitary operator
-$lambda(g)$ shifting a function: $(lambda(g)xi)(h)=xi(g^{-1}h)$. The
-group von Neumann algebra $L(Gamma)$ is the algebra of bounded
-operators on $ell^2(Gamma)$ generated by all these unitaries together
+follows. Let ell^2(Gamma) be the Hilbert space of square-summable
+functions on Gamma. For each ginGamma there is a unitary operator
+lambda(g) shifting a function: (lambda(g)xi)(h)=xi(g^{-1}h). The
+group von Neumann algebra L(Gamma) is the algebra of bounded
+operators on ell^2(Gamma) generated by all these unitaries together
with their (weak) limits.
An algebra of this kind remembers a surprising amount of the group. It
also forgets a surprising amount. The central question of the field, in
one form or another, is how much does the shadow remember? If two
-groups $Gamma$ and $Λ$ have isomorphic shadows $L(Gamma)cong
-L(Λ)$, must the groups be isomorphic? If the shadow retains, along
+groups Gamma and Λ have isomorphic shadows L(Gamma)cong
+L(Λ), must the groups be isomorphic? If the shadow retains, along
with any isomorphism, a controlled correspondence with the original
-group, the group is called $W^*$-superrigid.
+group, the group is called W^*-superrigid.
Recall the notion of a factor: a von Neumann algebra whose center
is trivial. A factor is the noncommutative analogue of an "indecomposable
system," a quantum object that cannot be split into independent
-subsystems. The condition that $Gamma$ be ICC is exactly what guarantees
-that $L(Gamma)$ is a factor of type $II_1$, the cleanest
+subsystems. The condition that Gamma be ICC is exactly what guarantees
+that L(Gamma) is a factor of type II_1, the cleanest
infinite-dimensional kind, on which a natural trace (a kind of
dimension function) is defined.
@@ -197,13 +197,13 @@ Sometimes the symmetries of a system are only defined up to a phase: a
rotation by the identity angle might act as multiplication by a complex
number of modulus one. Such an "almost action" is a projective
representation, and its failure of multiplicativity is encoded in a
-cocycle $mu:GammatimesGammatomathbb T$.bbT$. The rule is
+cocycle mu:GammatimesGammatomathbb T.bbT. The rule is
[
mu(g,h)mu(gh,j)=mu(h,j)mu(g,hj)
]
-with values in the unit circle $mathbb T$.bbT$. The twisted group von
-Neumann algebra $L_mu(Gamma)$ is built from $Gamma$ together with
-this cocycle: the generators $u_g$ now satisfy $u_gu_h=mu(g,h)u_{gh}$.
+with values in the unit circle mathbb T.bbT. The twisted group von
+Neumann algebra L_mu(Gamma) is built from Gamma together with
+this cocycle: the generators u_g now satisfy u_gu_h=mu(g,h)u_{gh}.
Scalar cocycles are unavoidable in the geometry of these problems, and a
central theme of this paper is that they must be kept, not discarded:
they appear as the exact ratio between the two endpoint twists of
@@ -211,10 +211,10 @@ the final models.
## 5. Finite correspondences: quantized relations
-A relation between two sets $X$ and $Y$ is a subset of $Xtimes Y$. In
-the quantum world, a relation between two von Neumann algebras $M$ and
-$N$ is an $M$--$N$ bimodule: a Hilbert space on which $M$ acts
-from the left and $N$ from the right, with the two actions commuting.
+A relation between two sets X and Y is a subset of Xtimes Y. In
+the quantum world, a relation between two von Neumann algebras M and
+N is an M--N bimodule: a Hilbert space on which M acts
+from the left and N from the right, with the two actions commuting.
This is the operator-algebraic analogue of a bridge or a coupling
between the two systems.
@@ -227,41 +227,41 @@ trace-preserving at both ends, the quantum analogue of a finite
relation.
The central question becomes: where can finite correspondences come
-from? One family of examples is always available. Suppose $Gamma$ and
-$Λ$ are groups and $AleqGamma$, $BleqΛ$ are subgroups of
-finite index, with an actual isomorphism $delta:Ato B$ between them.
+from? One family of examples is always available. Suppose Gamma and
+Λ are groups and AleqGamma, BleqΛ are subgroups of
+finite index, with an actual isomorphism delta:Ato B between them.
Suppose moreover there is a finite-dimensional projective unitary
-representation $sigma$ of $A$ whose multiplier matches the cocycle
+representation sigma of A whose multiplier matches the cocycle
ratio, meaning
[
sigma(a)sigma(b)=(mu(a,b))/(omega(delta(a),delta(b)))sigma(ab).
]
-Then there is a canonical finite correspondence $E(A,B,delta,sigma)$,
+Then there is a canonical finite correspondence E(A,B,delta,sigma),
given by a relative tensor product (Definition and construction in
Section~sec:models). These are the arithmetic models.
## 6. The question, and the answer
The question: given a finite correspondence between the twisted group
-factors $L_mu(Gamma)$ and $L_omega(Λ)$ for $Gammainmathscr
-K$scrK$ and arbitrary countably infinite ICC $Λ$, is it built out of
+factors L_mu(Gamma) and L_omega(Λ) for Gammainmathscr
+KscrK and arbitrary countably infinite ICC Λ, is it built out of
arithmetic models? The answer proved here is yes.
-Main theorem (Arithmetic exhaustion). Let $Gammainmathscr K$,scrK$,
-let $Λ$ be any countably infinite ICC group, and let $mu,omega$
+Main theorem (Arithmetic exhaustion). Let Gammainmathscr K,scrK,
+let Λ be any countably infinite ICC group, and let mu,omega
be normalized scalar cocycles. Every bifinite
-$L_mu(Gamma)$--$L_omega(Λ)$ correspondence is unitarily
+L_mu(Gamma)--L_omega(Λ) correspondence is unitarily
isomorphic to a closed bimodule summand of a finite direct sum of
-arithmetic models $E(A,B,delta,sigma)$, where $AleqGamma$ and
-$BleqΛ$ have finite index, $delta:Ato B$ is an actual group
-isomorphism, and $sigma$ is a finite-dimensional projective unitary
+arithmetic models E(A,B,delta,sigma), where AleqGamma and
+BleqΛ have finite index, delta:Ato B is an actual group
+isomorphism, and sigma is a finite-dimensional projective unitary
representation satisfying the matched-multiplier identity above.
Conversely, every such model, finite direct sum, and closed bimodule
summand is a normal bifinite correspondence. The elementary model has
-dimensions $n[Λ:B]$ on the left and $n[Gamma:A]$ on the right,
-where $n=dim V_sigma$.
+dimensions n[Λ:B] on the left and n[Gamma:A] on the right,
+where n=dim V_sigma.
Three things are worth stressing.
@@ -273,17 +273,17 @@ model; the summand and finite sum are necessary because a correspondence
can have irreducible pieces living in several distinct arithmetic models
at once.
-Second, only the matched ratio $mu/delta^*omega$ needs to admit a
+Second, only the matched ratio mu/delta^*omega needs to admit a
finite-dimensional projective representation. Neither endpoint cocycle
is required to have finite order or finite type, and the image of
-$sigma$ may be infinite. This is exactly where the twisting appears as an
+sigma may be infinite. This is exactly where the twisting appears as an
obstruction: existence of a nonzero correspondence of the simplest kind
-forces $Gamma$ and $Λ$ to be abstractly commensurable.
+forces Gamma and Λ to be abstractly commensurable.
Third, the theorem is unconditional within its class. It says nothing
-about general property-$(T)$ groups, and indeed property $(T)$ alone
+about general property-(T) groups, and indeed property (T) alone
fails to give group reconstruction: independent concurrent
-constructions give nonisomorphic ICC property-$(T)$ groups with
+constructions give nonisomorphic ICC property-(T) groups with
isomorphic group factors. What is special to our lattice class is the
geometric rigidity available inside the correspondence, and the
arithmetic exhaustion is the exact finite-correspondence input for the
@@ -291,15 +291,15 @@ stable recovery results of the companion article.
## 7. Why this is hard
-The comparison group $Λ$ is only assumed countably infinite
+The comparison group Λ is only assumed countably infinite
and ICC. Its von Neumann algebra is as wild as any such algebra can be.
-In particular we have no geometric action of $Λ$ on any symmetric
+In particular we have no geometric action of Λ on any symmetric
space or building: the comparison group is an algebraic cipher, not a
geometric object. Yet we must produce actual finite-index subgroup graphs
-of $Λ$. The only geometry available at the start is the geometry of
-$Gamma$ and its ambient groups $G_i$. The proof must manufacture
+of Λ. The only geometry available at the start is the geometry of
+Gamma and its ambient groups G_i. The proof must manufacture
geometry inside the correspondence from the source side alone, then
-must read $Λ$ through it.
+must read Λ through it.
The second hardness is scalar. Even in the classical, untwisted
case, the argument must be careful. With twists in the game, phases
@@ -315,7 +315,7 @@ read off the arithmetic from the geometry, working in small enough
pieces that phases stay visible. Here are the main steps.
Step 1. Reduce to a good lattice. First pass to an actual lattice
-$Gamma<G$ and decompose the finite commutant into compact projective
+Gamma<G and decompose the finite commutant into compact projective
sectors, discarding the compact parts. What remains is a correspondence
with weakly mixing conjugation action on the commutant of the
source factor. This is the clean hypothesis under which the geometric
@@ -344,17 +344,16 @@ show that the corresponding target Cartan projection grows at most
linearly in the source speed, with a comparison constant independent of
everything else. This is the upper speed theorem, and it is
proved separately at three kinds of local places:
-
-item Finite place (Section~sec:finite): compact-open fixed
-algebras, a uniform $p$-adic logarithm lemma, and a Hausdorff-limit
+ Finite place (Section~sec:finite): compact-open fixed
+algebras, a uniform p-adic logarithm lemma, and a Hausdorff-limit
argument on block stabilizers give the equality of source and target
-dimensions over the prime field and the upper bound $ell_T(g)le
-d_St+O(1)$.
-item Non-isolated archimedean place (Section~sec:panels):
+dimensions over the prime field and the upper bound ell_T(g)le
+d_St+O(1).
+ Non-isolated archimedean place (Section~sec:panels):
optimal finite-dimensional modules of the rank-one Levi factor, and
-$L^2$-trace inequalities on compact operators reconstructed from the
+L^2-trace inequalities on compact operators reconstructed from the
graph relation, give the analogous bound.
-item Isolated real place (Sections~sec:ray--sec:scale):
+ Isolated real place (Sections~sec:ray--sec:scale):
ray densities on the symmetric space, a uniform radial-multiplier
(completely bounded) estimate, and a "Kazhdan tightening" that pins down
a linear subsequence give, instead of an upper bound, a lower
@@ -365,8 +364,8 @@ tractable one.
Step 4. Build heights and calibrate. On the isolated factors, the
boundary measurements have a derivative in the horizontal direction,
-and the proof constructs a positive injective operator $b_x$ at each
-source label that commutes with the measurement $P_x$ and transforms
+and the proof constructs a positive injective operator b_x at each
+source label that commutes with the measurement P_x and transforms
under source and target translations by the inverse angular Jacobian.
This "height" is the analytic core of the isolated case
(Sections~sec:scale and~sec:heights). The height lets us
@@ -381,9 +380,9 @@ different labels must commute. This is not automatic; they are
a priori noncommuting operators. Section~sec:locality proves
commutation for all labels of an isolated factor by a deep
finite-dimensional locality argument: it builds a graph domain
-$mathfrak D$,frakD$, proves that the coefficients of an arbitrary connected
+mathfrak D,frakD, proves that the coefficients of an arbitrary connected
piece are finite-jets, uses the geometry of the boundary kernel
-$K(w,z)$ to eliminate positive differential order, and concludes that
+K(w,z) to eliminate positive differential order, and concludes that
the joint spectral law of the angle-height operators is exactly the law
of evaluation of a smooth homogeneous diffeomorphism
of the boundary. Then it proves that the two normal copies of the boundary
@@ -393,9 +392,9 @@ calculus commute as well. This is where the "quantum relation" becomes
Step 6. Glue. Do this at every local type—every finite-place panel,
every real panel, every isolated factor—and prove that the local maps can
be glued into one norm-bisecting, essentially invertible chamber map
-$B_Sto B_T$ (Sections~sec:panels and~sec:gluing). The
+B_Sto B_T (Sections~sec:panels and~sec:gluing). The
finite-place panels give homogeneous analytic maps with isomorphic
-$p$-adic differential; the real panels give smooth homogeneous maps
+p-adic differential; the real panels give smooth homogeneous maps
after a Taylor-density argument; the gluing sections install reflection
substitutions that propagate monotone commutativity along a longest Weyl
word.
@@ -408,7 +407,7 @@ correspondence, which are finite-dimensional). Prove that the atoms form
a deterministic, finite-rank partition, permuted by both endpoint
groups with finite orbits and a free source action. Then, by pure group
theory—no more boundary analysis—the joint stabilizer of one atom is the
-graph of a finite-index subgroup isomorphism $Ato B$, and the atom
+graph of a finite-index subgroup isomorphism Ato B, and the atom
carries the matched projective multiplier. This yields the arithmetic
models. Finally, undo the finite-index restriction and the compact
sector: everything restores and Theorem~thm:arithmetic-exhaustion
@@ -428,7 +427,7 @@ cocycles along the subgroup isomorphism.
The source groups are lattices in products of higher-rank real and
nonarchimedean groups, together with the quaternionic and Cayley
-property-$(T)$ rank-one groups. The comparison group is only assumed to be
+property-(T) rank-one groups. The comparison group is only assumed to be
countably infinite and ICC. Thus the geometry that the proof uses must be
constructed inside the correspondence: no boundary or ambient Lie
group is available on the comparison side at the start.
@@ -436,17 +435,17 @@ group is available on the comparison side at the start.
## The class, conventions, and main theorem
-Let $mathscr K$scrK$ be the class of countably infinite ICC groups abstractly
+Let mathscr KscrK be the class of countably infinite ICC groups abstractly
commensurable with a lattice in a finite nonempty product
[
- G=prod_{i=1}^r G_i,qquad G_i=mathbf H_i(k_i)^+.bfH_i(k_i)^+.
+ G=prod_{i=1}^r G_i, G_i=mathbf H_i(k_i)^+.bfH_i(k_i)^+.
]
-Here $k_i$ is $mathbb R$,bbR$, $mathbb C$,bbC$, or a finite extension of
-$mathbb Q_p$;bbQ_p$; the primes may differ. The algebraic group $mathbf H_i$bfH_i$ is
-connected, adjoint, and absolutely simple over $k_i$. The superscript $+$
-denotes the subgroup generated by the $k_i$-points of the unipotent
-radicals of proper $k_i$-parabolic subgroups. Each $G_i$ is required to be
-noncompact and to have Kazhdan's property $(T)$.
+Here k_i is mathbb R,bbR, mathbb C,bbC, or a finite extension of
+mathbb Q_p;bbQ_p; the primes may differ. The algebraic group mathbf H_ibfH_i is
+connected, adjoint, and absolutely simple over k_i. The superscript +
+denotes the subgroup generated by the k_i-points of the unipotent
+radicals of proper k_i-parabolic subgroups. Each G_i is required to be
+noncompact and to have Kazhdan's property (T).
A lattice is discrete and has finite Haar covolume. Abstract
commensurability means an actual isomorphism between finite-index
@@ -455,82 +454,82 @@ No irreducibility, cocompactness, product decomposition, or torsion
hypothesis is imposed on the lattice.
-Property $(T)$ means that every continuous unitary representation with
+Property (T) means that every continuous unitary representation with
almost invariant unit vectors admits a nonzero invariant vector. The real
factors in Definition~def:class are either higher rank or locally
-$Sp(n,1)$, $nge2$, or $F_{4(-20)}$; at a finite place the relative rank
+Sp(n,1), nge2, or F_{4(-20)}; at a finite place the relative rank
is at least two. Complex places are treated as real Lie groups when
differential arguments are used. Standard structure and permanence
results are recalled in Section~sec:setup; see also
[cite].
-All scalar cocycles are $mathbb T$-valuedbbT$-valued and normalized:
+All scalar cocycles are mathbb T-valuedbbT-valued and normalized:
[
- mu(g,h)mu(gh,j)=mu(h,j)mu(g,hj),qquad
+ mu(g,h)mu(gh,j)=mu(h,j)mu(g,hj),
mu(e,g)=mu(g,e)=1.
]
-The twisted group factor $L_mu(Gamma)$ has canonical unitaries $u_g$
-with $u_gu_h=mu(g,h)u_{gh}$ and trace $tau(u_g)=mathbf 1_bf1_{{e}}(g)$.
+The twisted group factor L_mu(Gamma) has canonical unitaries u_g
+with u_gu_h=mu(g,h)u_{gh} and trace tau(u_g)=mathbf 1_bf1_{{e}}(g).
For an ICC group this is a factor: the absolute values of Fourier
coefficients of a central vector are constant on conjugacy classes, so
square summability kills all nonidentity coefficients.
-A bifinite $M$--$N$ correspondence is a Hilbert space carrying
-commuting normal unital left $M$ and right $N$ actions and finite von
+A bifinite M--N correspondence is a Hilbert space carrying
+commuting normal unital left M and right N actions and finite von
Neumann module dimensions on both sides. The actions need not be mutual
commutants, and the correspondence need not be irreducible. Every
correspondence in this paper is separable. Every factor isomorphism is
-normal, unital, complex-linear, and $*$-preserving. For a finite factor
-$Q$ and $t>0$ we write
+normal, unital, complex-linear, and *-preserving. For a finite factor
+Q and t>0 we write
[
Q^t=p(M_n(mathbb C)bbC)barbar ⊗⊗ Q)p,
- qquad (Tr_n ⊗ tau_Q)(p)=t,
+ (Tr_n ⊗ tau_Q)(p)=t,
]
-with the unnormalized matrix trace $Tr_n$; the corner itself carries its
+with the unnormalized matrix trace Tr_n; the corner itself carries its
normalized trace.
-For finite-index subgroups $AleGamma$, $BleΛ$, a group
-isomorphism $delta:Ato B$, and a nonzero finite-dimensional projective
-unitary representation $sigma$ satisfying
+For finite-index subgroups AleGamma, BleΛ, a group
+isomorphism delta:Ato B, and a nonzero finite-dimensional projective
+unitary representation sigma satisfying
sigma(a)sigma(b)=(mu(a,b))/(omega(delta(a),delta(b)))sigma(ab),
- qquad sigma(e)=I,
+ sigma(e)=I,
define the arithmetic model
E(A,B,delta,sigma)
=L^2(M),barbar ⊗⊗ _{L_{mu|_A}(A)}
(V_sigma ⊗ L^2(N)),
- qquad M=L_mu(Gamma),quad N=L_omega(Λ).
+ M=L_mu(Gamma), N=L_omega(Λ).
-The right $N$-action is standard. The left subgroup action on the second
-factor is $u_a(xi ⊗ eta)=sigma(a)xi ⊗ v_{delta(a)}eta$;
+The right N-action is standard. The left subgroup action on the second
+factor is u_a(xi ⊗ eta)=sigma(a)xi ⊗ v_{delta(a)}eta;
Equation~(eq:intro-multiplier) makes this an action with multiplier
-$mu|_A$.
+mu|_A.
-A right action is represented by $rho(h)xi=xi v_h^*$, so
-$rho(h)rho(k)=omega(h,k)barrho(hk)$. The left action is
-$pi(g)xi=u_gxi$. Hence the joint action
-$U_{(g,h)}xi=u_gxi v_h^*$ has multiplier
-$alpha((g,h),(g',h'))=mu(g,g')omega(h,h')bar$.
+A right action is represented by rho(h)xi=xi v_h^*, so
+rho(h)rho(k)=omega(h,k)barrho(hk). The left action is
+pi(g)xi=u_gxi. Hence the joint action
+U_{(g,h)}xi=u_gxi v_h^* has multiplier
+alpha((g,h),(g',h'))=mu(g,g')omega(h,h')bar.
[Arithmetic exhaustion]
-Let $Gammainmathscr K$,scrK$, let $Λ$ be any countably infinite ICC
-group, and let $muin Z^2(Gamma,mathbb T)$,bbT)$,
-$omegain Z^2(Λ,mathbb T)$bbT)$ be normalized scalar cocycles.
-Every bifinite $L_mu(Gamma)$--$L_omega(Λ)$ correspondence is
+Let Gammainmathscr K,scrK, let Λ be any countably infinite ICC
+group, and let muin Z^2(Gamma,mathbb T),bbT),
+omegain Z^2(Λ,mathbb T)bbT) be normalized scalar cocycles.
+Every bifinite L_mu(Gamma)--L_omega(Λ) correspondence is
unitarily isomorphic to a closed bimodule summand of a finite direct sum
-of the models $E(A,B,delta,sigma)$ of Equation~(eq:intro-model),
-where $AleGamma$ and $BleΛ$ have finite index, $delta:Ato B$
-is an actual group isomorphism, and $sigma$ is a finite-dimensional
+of the models E(A,B,delta,sigma) of Equation~(eq:intro-model),
+where AleGamma and BleΛ have finite index, delta:Ato B
+is an actual group isomorphism, and sigma is a finite-dimensional
projective unitary representation satisfying
Equation~(eq:intro-multiplier).
Conversely, every such model, finite direct sum, and closed bimodule
summand is a normal bifinite correspondence. If
-$n=dim_{mathbb C}bbCV_sigma$, the elementary model has dimensions
+n=dim_{mathbb C}bbCV_sigma, the elementary model has dimensions
[
- dim_{L_mu(Gamma)}E(A,B,delta,sigma)=n[Λ:B],qquad
+ dim_{L_mu(Gamma)}E(A,B,delta,sigma)=n[Λ:B],
dim E(A,B,delta,sigma)_{L_omega(Λ)}=n[Gamma:A].
]
@@ -545,12 +544,12 @@ Theorem~thm:arithmetic-exhaustion as its geometric input. The
elementary construction and its normality and dimensions are proved here
in Section~sec:models.
-Only the matched ratio $mu/delta^*omega$ must admit the
+Only the matched ratio mu/delta^*omega must admit the
finite-dimensional projective representation of
Equation~(eq:intro-multiplier). Neither endpoint cocycle is
required to have finite order, finite type, or a virtually trivial
-restriction, and the image of $sigma$ may be infinite. In particular,
-existence of a nonzero correspondence forces $Gamma$ and $Λ$ to be
+restriction, and the image of sigma may be infinite. In particular,
+existence of a nonzero correspondence forces Gamma and Λ to be
abstractly commensurable. These conclusions cover nonuniform and
reducible lattices and their ICC commensurable groups.
@@ -558,17 +557,17 @@ reducible lattices and their ICC commensurable groups.
The group-factor reconstruction problem asks which features of a discrete
group survive in its von Neumann algebra. Connes proved countability of
-the fundamental group of a property-$(T)$ group factor [cite];
-his later problems ask whether nonisomorphic ICC property-$(T)$ groups
+the fundamental group of a property-(T) group factor [cite];
+his later problems ask whether nonisomorphic ICC property-(T) groups
have nonisomorphic factors, and ask for the fundamental groups of those
-factors cite[Chapter~V, Appendix~B.$ε$, Problems~1--2]{Connes1994}.
+factors cite[Chapter~V, Appendix~B.ε, Problems~1--2]{Connes1994}.
Popa's survey states a stronger recovery question that controls the given
isomorphism, up to a character and inner conjugacy, together with its
-amplification scale cite[Section~3, Equations~(3.2) and~(3.2$'$)]{Popa2007}.
+amplification scale cite[Section~3, Equations~(3.2) and~(3.2')]{Popa2007}.
-Property $(T)$ alone does not ensure group reconstruction: independent,
+Property (T) alone does not ensure group reconstruction: independent,
concurrent constructions of OpenAI and Zhou give nonisomorphic ICC
-property-$(T)$ groups with isomorphic group factors
+property-(T) groups with isomorphic group factors
cite[Theorem~1.2]{OpenAI2026ConnesCounterexample}
cite[Theorem~A]{Zhou2026Counterexamples}. The present theorem concerns
the lattice class of Definition~def:class. Arithmetic exhaustion is
@@ -577,10 +576,10 @@ stable canonical recovery for this class.
Deformation and rigidity methods have produced strong reconstruction
theorems in several settings. Ioana's theorem recovers Bernoulli actions
-of ICC property-$(T)$ groups from their crossed products
-[cite]. Ioana--Popa--Vaes obtained $W^*$-superrigid groups among
+of ICC property-(T) groups from their crossed products
+[cite]. Ioana--Popa--Vaes obtained W^*-superrigid groups among
generalized wreath products [cite], and
-Chifan--Ioana--Osin--Sun constructed property-$(T)$ examples using
+Chifan--Ioana--Osin--Sun constructed property-(T) examples using
wreath-like products [cite]. These results recover groups or
actions from operator-algebraic data through constructions different from
the lattice geometry developed here.
@@ -599,16 +598,16 @@ left-right wreath products, including the projective-ratio constraint and
reconstruction, up to amplification, of finite-index factor neighbours
cite[Theorems~A and~6.6]{DonvilVaes2025}.
Chifan--Fern'andez Quero--Osin--Tan prove stable canonical recovery for
-twists of certain property-$(T)$ wreath-like products when the source
+twists of certain property-(T) wreath-like products when the source
cocycle is trivial on the abelian base cite[Theorem~F]{CFOT2026}. The
exhaustion theorem proved here treats arbitrary source cocycles in the
local-field lattice class.
A geometric line of operator-algebraic rigidity for lattices begins with
character and finite-factor representation results. Bekka classified the
-characters of $SL_n(mathbb Z)$,bbZ)$, $nge3$ cite[Theorem~3]{Bekka2007};
+characters of SL_n(mathbb Z),bbZ), nge3 cite[Theorem~3]{Bekka2007};
Peterson established a finite-factor representation dichotomy for
-irreducible higher-rank property-$(T)$ lattices
+irreducible higher-rank property-(T) lattices
cite[Theorem~A]{Peterson2014}. Boutonnet--Houdayer developed stationary
and noncommutative boundary methods that recover character rigidity
cite[Theorems~A--C]{BoutonnetHoudayer2021}. Their noncommutative factor
@@ -618,7 +617,7 @@ quotients cite[Theorem~B and Theorem~3.3]{BoutonnetHoudayer2023}.
Related results recover Weyl groups from the distinguished inclusion
associated with the action on the quotient by a split-torus centralizer
cite[Theorem~A and Corollary~B]{HI2026}, and classify intermediate
-$C^*$-algebras for boundary crossed products of irreducible lattices in
+C^*-algebras for boundary crossed products of irreducible lattices in
products of real rank-one groups cite[Corollary~B]{HoudayerLeBars2026}.
These results show how boundary data retains the ambient geometry. For
the finite correspondence considered here, compatible boundary
@@ -637,29 +636,29 @@ consequences of that classical theorem.
The proof first passes to an actual lattice and removes the compact
projective sectors of the finite commutant. The remaining conjugation
action is weakly mixing. In this reduced setting the goal is a fixed
-orthogonal partition of the correspondence $K$ into finite-dimensional
+orthogonal partition of the correspondence K into finite-dimensional
subspaces permuted by both endpoint groups, with free actions and finitely
many orbits. A joint stabilizer of one subspace is then the graph of an
actual finite-index subgroup isomorphism, and its action on that subspace
is the projective fibre in Equation~(eq:intro-multiplier).
The geometric construction produces this partition through boundary
-measurements. Let $B$ be the compact full flag space of the known ambient
-group $G$. A projection-valued measure (PVM) assigns an orthogonal
+measurements. Let B be the compact full flag space of the known ambient
+group G. A projection-valued measure (PVM) assigns an orthogonal
projection to each Borel set, with countable additivity in the strong
operator topology; disjoint sets give orthogonal projections and the whole
-space gives the identity. An angular measurement $P_x$ on $B$ records the
-possible image of a source flag $xin B$: $P_x(A)$ is the projection for
-its image to lie in $A⊆ B$. Initially the construction supplies
+space gives the identity. An angular measurement P_x on B records the
+possible image of a source flag xin B: P_x(A) is the projection for
+its image to lie in A⊆ B. Initially the construction supplies
positive operator-valued observations. Showing that they are PVMs, and
that observations at different labels commute, are separate steps.
-To compare two source copies, use $K_0 ⊗ K_1$, with the left
-$Gamma$-action in each slot and the right $Λ$-action acting
-diagonally. The eventual joint angular law is a PVM $Θ$ on the space
-of nonsingular, essentially invertible maps $Bto B$. The two left actions
-transport its source and target labels; diagonal $rho(Λ)$ fixes
-it. The same construction on $K_0 ⊗ K_1 ⊗ K_2$ compares slot
+To compare two source copies, use K_0 ⊗ K_1, with the left
+Gamma-action in each slot and the right Λ-action acting
+diagonally. The eventual joint angular law is a PVM Θ on the space
+of nonsingular, essentially invertible maps Bto B. The two left actions
+transport its source and target labels; diagonal rho(Λ) fixes
+it. The same construction on K_0 ⊗ K_1 ⊗ K_2 compares slot
zero separately with slots one and two, and the required conclusion
includes
[
@@ -667,19 +666,19 @@ includes
qquadfor all Borel sets of boundary maps A,C.
]
This common-slot commutation has a concrete purpose. Normal slices in the
-separate branch slots generate an abelian algebra on $K_0$. Disintegrating
-over it gives boundary-map PVMs on the fixed space $K_1$. Finite module
+separate branch slots generate an abelian algebra on K_0. Disintegrating
+over it gives boundary-map PVMs on the fixed space K_1. Finite module
dimension makes their orbit transversals finite rank; weak mixing makes
the resulting projection partition independent of the disintegration
parameter. The boundary-map labels themselves need not become
deterministic. Section~sec:recovery proves this passage and the
subsequent graph-stabilizer identification.
-Both copies of $B$ in this construction belong to the source lattice. No
-flag space for the unknown group $Λ$ is assumed. To read the known
+Both copies of B in this construction belong to the source lattice. No
+flag space for the unknown group Λ is assumed. To read the known
geometry from Fourier coefficients, omit one left action and put the
-remaining actions, together with diagonal $rho$, in a finite regular
-column. Inducing the omitted action to $G/Gamma$ gives a finite tracial
+remaining actions, together with diagonal rho, in a finite regular
+column. Inducing the omitted action to G/Gamma gives a finite tracial
algebra even when the lattice is reducible. Escape in all ambient factors
then creates an isometric boundary link. Escape in a subproduct instead
gives a pairing through the fixed algebra's conditional expectation.
@@ -764,25 +763,25 @@ cite[Section~4]{Mackey1958}, and these counting-space coordinates also
belong to the finite-index bimodule calculus of
cite[Section~6.2]{Vaes2008}.
-Throughout, $Gamma,Λ$ are arbitrary countably infinite ICC
-groups, $M=L_mu(Gamma)$, $N=L_omega(Λ)$, and
+Throughout, Gamma,Λ are arbitrary countably infinite ICC
+groups, M=L_mu(Gamma), N=L_omega(Λ), and
[
- D=GammatimesΛ,qquad
+ D=GammatimesΛ,
alpha((g,h),(g',h'))=mu(g,g')omega(h,h')bar.
]
-For a subgroup $Cle D$ the graph of an isomorphism
-$delta:Ato B$ of finite-index subgroups is
-$C_delta={(a,delta(a)):ain A}le D$.
+For a subgroup Cle D the graph of an isomorphism
+delta:Ato B of finite-index subgroups is
+C_delta={(a,delta(a)):ain A}le D.
## Projective induction
-Let $Cle D$ be a subgroup and let $gamma$ be a normalized unitary
-projective representation on a finite-dimensional space $V$, with
-multiplier $alpha|_{Ctimes C}$. Write $X=D/C$ for the left cosets,
-choose representatives $r_xin D$ with $r_C=e$, and give each $xin X$ a
-copy $V_x$ of $V$. Set
-$mathcal H(C,calH(C,gamma)=⊕_{xin X}V_x$, with the counting Hilbert
-norm. For $sin D$ put
+Let Cle D be a subgroup and let gamma be a normalized unitary
+projective representation on a finite-dimensional space V, with
+multiplier alpha|_{Ctimes C}. Write X=D/C for the left cosets,
+choose representatives r_xin D with r_C=e, and give each xin X a
+copy V_x of V. Set
+mathcal H(C,calH(C,gamma)=⊕_{xin X}V_x, with the counting Hilbert
+norm. For sin D put
c(s,x)&=r_{sx}^{-1}sr_x,
@@ -795,59 +794,59 @@ norm. For $sin D$ put
[Exact projective induction]
Equation~(eq:input-A1) defines a unitary projective representation
-of $D$ with multiplier $alpha$. Changing the coset representatives gives
+of D with multiplier alpha. Changing the coset representatives gives
a unitarily equivalent bundle by unitary identifications of its
-individual fibres. If $C$ is the graph of a finite-index subgroup
-isomorphism $Ato B$, the bundle has commuting normal $M$ and right $N$
+individual fibres. If C is the graph of a finite-index subgroup
+isomorphism Ato B, the bundle has commuting normal M and right N
actions, and is bifinite.
Proof.
-The correction $c(s,x)$ belongs to $C$, and
+The correction c(s,x) belongs to C, and
[
c(s,tx)c(t,x)=c(st,x).
]
-Using the cocycle identity for $alpha$ together with the projective law
-of $gamma$ in Equation~(eq:input-A1) gives
+Using the cocycle identity for alpha together with the projective law
+of gamma in Equation~(eq:input-A1) gives
U^X_s(tx)U^X_t(x)=alpha(s,t)U^X_{st}(x).
Equivalently, all scalar factors can be verified at once in the central
-extension with multiplication $(z,s)(w,t)=(zwalpha(s,t),st)$: the
-representation of the preimage of $C$ is $(z,c)mapsto zgamma(c)$, and
-$sr_x=r_{sx}c(s,x)$ gives exactly the ratio in Equation~(eq:input-A1).
+extension with multiplication (z,s)(w,t)=(zwalpha(s,t),st): the
+representation of the preimage of C is (z,c)mapsto zgamma(c), and
+sr_x=r_{sx}c(s,x) gives exactly the ratio in Equation~(eq:input-A1).
This argument retains the multiplier rather than discarding it.
For explicit independence of the choice of representatives, suppose
-$r'_x=r_xk_x$ with $k_xin C$. The identification from the new
-coordinate copy of $V_x$ to the old one is
+r'_x=r_xk_x with k_xin C. The identification from the new
+coordinate copy of V_x to the old one is
[
F_x=alpha(r_x,k_x)^{-1}gamma(k_x).
]
-Indeed, identifying a fibre via $U_{r'_x}$ instead of $U_{r_x}$ has
+Indeed, identifying a fibre via U_{r'_x} instead of U_{r_x} has
precisely this effect, and Equation~(eq:calculus-transport-identity)
-then gives $F_{sx}U'{}^X_s(x)=U^X_s(x)F_x$.
+then gives F_{sx}U'{}^X_s(x)=U^X_s(x)F_x.
-Now assume that $C$ is a graph $C_delta$. Both coordinate groups act
-freely on $X$: if $(g,e)r_xC=r_xC$, then $r_x^{-1}(g,e)r_xin C$ has
-second coordinate $e$, hence $g=e$. On an orbit of the first coordinate
-group choose a fibre $V_{x_0}$. The unitary identification
+Now assume that C is a graph C_delta. Both coordinate groups act
+freely on X: if (g,e)r_xC=r_xC, then r_x^{-1}(g,e)r_xin C has
+second coordinate e, hence g=e. On an orbit of the first coordinate
+group choose a fibre V_{x_0}. The unitary identification
[
ell^2(Gamma) ⊗ V_{x_0}→
⊕_{ginGamma}V_{(g,e)x_0},
- qquad delta_g ⊗ xi⟼ U^X_{(g,e)}xi
+ delta_g ⊗ xi⟼ U^X_{(g,e)}xi
]
-intertwines the action with $lambda_mu ⊗ 1$, because
-$U_{(a,e)}U_{(g,e)}=mu(a,g)U_{(ag,e)}$. Hence it extends normally to
-$L_mu(Gamma)$. The same argument for the second coordinate gives copies
-of $lambda_{overlineomega}$, which are the standard normal
-representations of the right generators $ximapstoxi v_h^*$. The two
+intertwines the action with lambda_mu ⊗ 1, because
+U_{(a,e)}U_{(g,e)}=mu(a,g)U_{(ag,e)}. Hence it extends normally to
+L_mu(Gamma). The same argument for the second coordinate gives copies
+of lambda_{overlineomega}, which are the standard normal
+representations of the right generators ximapstoxi v_h^*. The two
actions commute, since the product cocycle has trivial mixed-coordinate
factors.
-The first-coordinate orbits are indexed by $Λ/B$ and the
-second-coordinate orbits by $Gamma/A$; both are finite. Each orbit is a
-regular module with multiplicity $dim_{mathbb C}bbCV$, proving bifiniteness
+The first-coordinate orbits are indexed by Λ/B and the
+second-coordinate orbits by Gamma/A; both are finite. Each orbit is a
+regular module with multiplicity dim_{mathbb C}bbCV, proving bifiniteness
and the dimension formulas below.
@@ -858,49 +857,49 @@ the model.
[The graph-bundle unitary]
-Let $delta:Ato B$ be a finite-index subgroup isomorphism and let
-$sigma:Atomathcal U(V)$calU(V)$ have multiplier
-$beta=mu|_{Atimes A}/delta^*omega$. Set
-$gamma(a,delta(a))=sigma(a)$. Then
+Let delta:Ato B be a finite-index subgroup isomorphism and let
+sigma:Atomathcal U(V)calU(V) have multiplier
+beta=mu|_{Atimes A}/delta^*omega. Set
+gamma(a,delta(a))=sigma(a). Then
[
E(A,B,delta,sigma)=
L^2(M)mathbin{barbar ⊗⊗ }_{L_{mu|A}(A)}
(V ⊗ L^2(N))
congmathcal H(C_calH(C_delta,gamma).
]
-The unitary sends the vector $xi$ in the fibre labelled $x$ to
-$U_{r_x}jxi$, where
-$jxi=widehatwidehat11 ⊗ (xi ⊗ widehatwidehat1)$1)$ in the displayed model.
+The unitary sends the vector xi in the fibre labelled x to
+U_{r_x}jxi, where
+jxi=widehatwidehat11 ⊗ (xi ⊗ widehatwidehat1)1) in the displayed model.
Proof.
-On $V ⊗ L^2(N)$ the subgroup action is
+On V ⊗ L^2(N) the subgroup action is
[
u_a(xi ⊗ eta)=sigma(a)xi ⊗ v_{delta(a)}eta.
]
-Its multiplier is $beta(a,b)omega(delta(a),delta(b))=mu(a,b)$. For
-normality, choose representatives $t$ for $BbackslashΛ$. On the
-summand spanned by $xi ⊗ v_{delta(a)}v_t$, the map
+Its multiplier is beta(a,b)omega(delta(a),delta(b))=mu(a,b). For
+normality, choose representatives t for BbackslashΛ. On the
+summand spanned by xi ⊗ v_{delta(a)}v_t, the map
xi ⊗ v_{delta(a)}v_t
⟼delta_a ⊗ sigma(a)^*xi
quadin ell^2(A) ⊗ V
is unitary and conjugates the subgroup action to
-$lambda_{mu|A} ⊗ 1$. To check the phase, applying $u_b$ before this
+lambda_{mu|A} ⊗ 1. To check the phase, applying u_b before this
map produces
[
omega(delta(b),delta(a))sigma(ba)^*sigma(b)
=omega(delta(b),delta(a))beta(b,a)sigma(a)^*
=mu(b,a)sigma(a)^*.
]
-Thus the left subgroup action is normal. The right $N$-action is standard
+Thus the left subgroup action is normal. The right N-action is standard
on the second factor and commutes with it.
-Choose representatives $r$ for $Gamma/A$. As a right
-$L_{mu|A}(A)$-module, $L^2(M)$ has orthonormal free basis $u_r$; the
+Choose representatives r for Gamma/A. As a right
+L_{mu|A}(A)-module, L^2(M) has orthonormal free basis u_r; the
assertion follows directly from
-$E_{L_{mu|A}(A)}(u_r^*u_{r'})=delta_{r,r'}1$. The relative product
+E_{L_{mu|A}(A)}(u_r^*u_{r'})=delta_{r,r'}1. The relative product
therefore identifies with the orthogonal sum
E(A,B,delta,sigma)
@@ -912,20 +911,20 @@ At its identity fibre the graph action is exactly
=u_a ⊗ (xi ⊗ v_{delta(a)}^*)
=jsigma(a)xi.
]
-For $g=ra$, balancing in Equation~(eq:calculus-coset-model) gives
+For g=ra, balancing in Equation~(eq:calculus-coset-model) gives
the more explicit formula
[
U_{(g,h)}jxi
=mu(r,a)^{-1}u_r ⊗
(sigma(a)xi ⊗ v_{delta(a)}v_h^*).
]
-The last Fourier coordinate is $delta(a)h^{-1}$, up to its displayed
+The last Fourier coordinate is delta(a)h^{-1}, up to its displayed
unit scalar. Two translated identity fibres are therefore orthogonal
-unless their labels in $D/C_delta$ agree: equality of the first cosets
+unless their labels in D/C_delta agree: equality of the first cosets
and the last Fourier coordinates is exactly that coset equality.
Conversely, these fibres exhaust every summand in
Equation~(eq:calculus-coset-model): they form an orthogonal,
-transitive system with graph action $gamma$ at the basepoint. Finally
+transitive system with graph action gamma at the basepoint. Finally
[
U_sU_{r_x}j
=frac{alpha(s,r_x)}{alpha(r_{sx},c(s,x))}
@@ -936,18 +935,18 @@ Equation~(eq:input-A1).
[Elementary normality and dimensions]
-The correspondence $E(A,B,delta,sigma)$ is normal and bifinite. If
-$n=dim_{mathbb C}bbCV_sigma$, then
+The correspondence E(A,B,delta,sigma) is normal and bifinite. If
+n=dim_{mathbb C}bbCV_sigma, then
[
- dim_M E(A,B,delta,sigma)=n[Λ:B],qquad
+ dim_M E(A,B,delta,sigma)=n[Λ:B],
dim E(A,B,delta,sigma)_N=n[Gamma:A].
]
Proof.
Use Lemma~lem:models-graph together with the orbit decompositions of
-Lemma~lem:models-induction. Each free coordinate orbit gives $n$
-regular copies, of module dimension $n$ in the unnormalized matrix-trace
-convention. The respective orbit counts are $[Λ:B]$ and $[Gamma:A]$.
+Lemma~lem:models-induction. Each free coordinate orbit gives n
+regular copies, of module dimension n in the unnormalized matrix-trace
+convention. The respective orbit counts are [Λ:B] and [Gamma:A].
Finite direct sums and closed bimodule summands of these models remain
@@ -973,14 +972,14 @@ Conjugation by these transports gives a genuine action of its ambient
group on the finite algebra. The remaining subsections describe the
geometric data to be read from the source-lattice Fourier indices.
-Fix a nonzero bifinite $L_mu(Gamma)$--$L_omega(Λ)$
-correspondence $K$ as in Theorem~thm:arithmetic-exhaustion. A right
-action of $L_omega(Λ)$ is written as the projective representation
+Fix a nonzero bifinite L_mu(Gamma)--L_omega(Λ)
+correspondence K as in Theorem~thm:arithmetic-exhaustion. A right
+action of L_omega(Λ) is written as the projective representation
[
- rho(h)xi=xi v_h^*,qquad
+ rho(h)xi=xi v_h^*,
rho(h)rho(k)=omega(h,k)barrho(hk).
]
-The left representation is $pi(g)xi=u_gxi$. Thus $pi$ and $rho$
+The left representation is pi(g)xi=u_gxi. Thus pi and rho
commute, although their multipliers need not be trivial. Conjugation by
either representation is an ordinary group action. All Hilbert spaces and
von Neumann algebras below are separable in the appropriate sense for
@@ -989,7 +988,7 @@ their normal representations.
## Compact reduction and finite-index return
We will prove the geometric reconstruction first for an actual ICC
-lattice $Gamma<G$ and under the condition that
+lattice Gamma<G and under the condition that
Adrho on
L^2(pi(L_mu(Gamma))',tr)
@@ -1004,20 +1003,20 @@ Section~sec:recovery.
[Twisted regular absorption]
-Let $alpha,nu$ be normalized scalar cocycles on a countable group $L$,
-and let $v$ be a $nu$-projective unitary representation on a Hilbert
-space $V$. Then
+Let alpha,nu be normalized scalar cocycles on a countable group L,
+and let v be a nu-projective unitary representation on a Hilbert
+space V. Then
[
lambda_alpha ⊗ vsimeqlambda_{alphanu} ⊗ 1_V.
]
-If $L$ is ICC and $V$ is finite-dimensional, tensoring a finite normal
-$L_alpha(L)$-module with $v$ gives a finite normal
-$L_{alphanu}(L)$-module. Any operators commuting with every $v_h$ are
+If L is ICC and V is finite-dimensional, tensoring a finite normal
+L_alpha(L)-module with v gives a finite normal
+L_{alphanu}(L)-module. Any operators commuting with every v_h are
preserved by the absorption identification.
Proof.
-On $ell^2(L) ⊗ V$ use the unitary
+On ell^2(L) ⊗ V use the unitary
[
T(delta_g ⊗ xi)=delta_g ⊗ v_g^*xi.
]
@@ -1028,104 +1027,104 @@ Indeed
]
A finite normal module over a finite factor embeds in finitely many copies
of the regular module cite[Theorems~10.1.1 and~10.2.1]{Jones2009}. When
-$L$ is ICC both twisted group algebras are factors, so applying this
+L is ICC both twisted group algebras are factors, so applying this
unitary to such an embedding proves the second assertion. The same
formula also proves the last assertion directly.
[Compact commutant reduction]
-Let $K$ be a nonzero bifinite
-$L_mu(Gamma)$--$L_omega(Λ)$ correspondence with both groups ICC.
-There is a finite-index subgroup $Λ_0leΛ$ such that the
-restriction of $K$ to the right subgroup factor is a finite orthogonal
+Let K be a nonzero bifinite
+L_mu(Gamma)--L_omega(Λ) correspondence with both groups ICC.
+There is a finite-index subgroup Λ_0leΛ such that the
+restriction of K to the right subgroup factor is a finite orthogonal
sum of blocks
[
- K_c=V_c ⊗ K'_c,qquad
- pi(g)|_{K_c}=1 ⊗ pi'_c(g),qquad
- rho(h)|_{K_c}=v^c_h ⊗ rho'_c(h)quad(hinΛ_0).
+ K_c=V_c ⊗ K'_c,
+ pi(g)|_{K_c}=1 ⊗ pi'_c(g),
+ rho(h)|_{K_c}=v^c_h ⊗ rho'_c(h)(hinΛ_0).
]
-The spaces $V_c$ are finite-dimensional. The $v^c_h$ are normalized
-projective matrix representations with multipliers $nu_c$, and
+The spaces V_c are finite-dimensional. The v^c_h are normalized
+projective matrix representations with multipliers nu_c, and
rho'_c(h)rho'_c(k)
=frac{omega(h,k)bar}{nu_c(h,k)}rho'_c(hk).
-Each $K'_c$ is a finite normal module on both sides and satisfies
+Each K'_c is a finite normal module on both sides and satisfies
Equation~(eq:setup-weak-mixing) for its reduced representations.
-No assertion is made that any $v^c$ has finite image.
+No assertion is made that any v^c has finite image.
Proof.
-Put $M=L_mu(Gamma)$, $N=L_omega(Λ)$, and $P=pi(M)'$. Finite
-$M$-module dimension makes $P$ a finite factor, and $N^{op}subset
-P$ has finite index. Indeed, regard $K$ as the equivalence module for the
-commuting factors $pi(M)$ and $P$; it has finite nonzero $P$-dimension,
-so finitely many copies of it contain $L^2(P)$ as a $P$-module. Their
-restrictions to $N^{op}$ are finite by right finiteness of $K$.
-Hence $L^2(P)$ has finite $N^{op}$-dimension, which is the index
+Put M=L_mu(Gamma), N=L_omega(Λ), and P=pi(M)'. Finite
+M-module dimension makes P a finite factor, and N^{op}subset
+P has finite index. Indeed, regard K as the equivalence module for the
+commuting factors pi(M) and P; it has finite nonzero P-dimension,
+so finitely many copies of it contain L^2(P) as a P-module. Their
+restrictions to N^{op} are finite by right finiteness of K.
+Hence L^2(P) has finite N^{op}-dimension, which is the index
assertion [cite].
-Let $D_csubset P$ be the bounded elements whose $Adrho$-orbits are
-precompact in $L^2(P)$. Compact vectors of a unitary representation form
+Let D_csubset P be the bounded elements whose Adrho-orbits are
+precompact in L^2(P). Compact vectors of a unitary representation form
its closed almost-periodic subspace. Bounded compact elements are closed
under adjoints and products, since on uniformly bounded sets
[
|xy-x'y'|_2le|x|_∞|y-y'|_2+|x-x'|_2|y'|_∞,
]
and their bounded balls are strongly closed because bounded strong
-convergence in a finite von Neumann algebra implies $L^2$ convergence.
-They are dense in the compact Hilbert subspace: the $L^2$-nearest-point
+convergence in a finite von Neumann algebra implies L^2 convergence.
+They are dense in the compact Hilbert subspace: the L^2-nearest-point
projection onto each closed operator-norm ball is equivariant and
-nonexpansive. Thus $D_c$ is a von Neumann subalgebra.
+nonexpansive. Thus D_c is a von Neumann subalgebra.
-Twisted conjugation on $L^2(N)ominusmathbb C1$bbC1$ has no nonzero
+Twisted conjugation on L^2(N)ominusmathbb C1bbC1 has no nonzero
finite-dimensional subrepresentation: the diagonal of a finite-rank
invariant projection in the Fourier basis is constant on conjugacy
classes, scalar phases disappear from the diagonal, the diagonal is
summable, and every nonidentity class is infinite, so only the identity
vector remains. The trace-preserving expectation
-$E_{N^{op}}:Pto N^{op}$ is equivariant and
-$L^2$-contractive, hence $E_{N^{op}}(x)=tr(x)1$
-for $xin D_c$. By the finite-index inequality there is $ε>0$
-with $E_{N^{op}}(x)geε x$ for $xge0$, so a nonzero
-projection $qin D_c$ has $tr(q)geε$. A finite
+E_{N^{op}}:Pto N^{op} is equivariant and
+L^2-contractive, hence E_{N^{op}}(x)=tr(x)1
+for xin D_c. By the finite-index inequality there is ε>0
+with E_{N^{op}}(x)geε x for xge0, so a nonzero
+projection qin D_c has tr(q)geε. A finite
algebra with this lower bound has only finitely many orthogonal nonzero
projections, and is finite-dimensional.
-Restrict to the finite-index subgroup $Λ_0$ fixing all central
-blocks of $D_c$. A central block $z$ reduces both endpoint actions; write
-$zK=V_c ⊗ K'_c$ and $zD_c=mathcal B(V_c)calB(V_c) ⊗ 1$. The left action
-commutes with $D_c$, hence is $1 ⊗ pi'_c$. Choose normalized matrix
-implementers $v^c_h$ for $Adrho(h)|_{zD_c}$; uniqueness up to a scalar
-gives $v^c_hv^c_k=nu_c(h,k)v^c_{hk}$, and after removing these
-implementers $rho(h)$ lies in the commutant of $mathcal B(V_c)$,calB(V_c)$, giving
+Restrict to the finite-index subgroup Λ_0 fixing all central
+blocks of D_c. A central block z reduces both endpoint actions; write
+zK=V_c ⊗ K'_c and zD_c=mathcal B(V_c)calB(V_c) ⊗ 1. The left action
+commutes with D_c, hence is 1 ⊗ pi'_c. Choose normalized matrix
+implementers v^c_h for Adrho(h)|_{zD_c}; uniqueness up to a scalar
+gives v^c_hv^c_k=nu_c(h,k)v^c_{hk}, and after removing these
+implementers rho(h) lies in the commutant of mathcal B(V_c),calB(V_c), giving
the asserted tensor form and Equation~(eq:setup-reduced-right-multiplier).
-Normality and finiteness of $pi'_c$ follow by taking a matrix corner of
-the finite normal left module. For $rho'_c$, first embed the original
-restricted $rho$ in a finite multiple of the
-$overlineomega|_{Λ_0}$-regular representation, possible because
+Normality and finiteness of pi'_c follow by taking a matrix corner of
+the finite normal left module. For rho'_c, first embed the original
+restricted rho in a finite multiple of the
+overlineomega|_{Λ_0}-regular representation, possible because
finite-index restriction preserves finite module dimension. Tensor with
-$v^cbar$; Lemma~lem:setup-twisted-absorption gives a finite
-normal module with multiplier $nu_cbar,overlineomega$. The
+v^cbar; Lemma~lem:setup-twisted-absorption gives a finite
+normal module with multiplier nu_cbar,overlineomega. The
vector corresponding to the identity matrix is fixed by
-$v^c_hbar ⊗ v^c_h$, so tensoring that line with $K'_c$
-exhibits $rho'_c$ as a summand of a normal finite representation with the
+v^c_hbar ⊗ v^c_h, so tensoring that line with K'_c
+exhibits rho'_c as a summand of a normal finite representation with the
multiplier of Equation~(eq:setup-reduced-right-multiplier).
Finally, a compact vector in the reduced commutant embeds as
-$1_{V_c} ⊗ xi$ on $zK$ and as zero on the other blocks, and is
-compact for $Λ_0$ hence for $Λ$; it therefore lies in
-$L^2(D_c)$ and its intersection with the reduced commutant on this block
+1_{V_c} ⊗ xi on zK and as zero on the other blocks, and is
+compact for Λ_0 hence for Λ; it therefore lies in
+L^2(D_c) and its intersection with the reduced commutant on this block
is scalar. This proves Equation~(eq:setup-weak-mixing).
[Finite-index restriction and return]
-Let $Gamma_0leGamma$ and $Λ_0leΛ$ have finite index, and
-restrict both cocycles literally. Every $M$--$N$ correspondence is a
+Let Gamma_0leGamma and Λ_0leΛ have finite index, and
+restrict both cocycles literally. Every M--N correspondence is a
summand of the two-sided induction of its restriction. The inclusions are,
on the respective sides,
@@ -1144,19 +1143,19 @@ on finite-index restriction.
Proof.
-The $u_r$ in Equation~(eq:setup-left-induction) are an orthonormal
-right basis for $L^2(M)$ over the subgroup factor, since
-$E_{L_{mu|Gamma_0}(Gamma_0)}(u_r^*u_{r'})=delta_{r,r'}1$; the
-analogous left-basis identity for the $v_s$ holds. The relative
+The u_r in Equation~(eq:setup-left-induction) are an orthonormal
+right basis for L^2(M) over the subgroup factor, since
+E_{L_{mu|Gamma_0}(Gamma_0)}(u_r^*u_{r'})=delta_{r,r'}1; the
+analogous left-basis identity for the v_s holds. The relative
tensor-product inner products give the isometries.
-For left equivariance, if $gr=r'a$ with $ainGamma_0$, then
-$u_gu_r=(mu(g,r)/mu(r',a))u_{r'}u_a$; moving $u_a$ across the balanced
-tensor product shows each term is the $r'$-term of $J_L(pi(u_g)xi)$. The
-right calculation is the same in the opposite algebra: when $sh=at$ with
-$ainΛ_0$,
+For left equivariance, if gr=r'a with ainGamma_0, then
+u_gu_r=(mu(g,r)/mu(r',a))u_{r'}u_a; moving u_a across the balanced
+tensor product shows each term is the r'-term of J_L(pi(u_g)xi). The
+right calculation is the same in the opposite algebra: when sh=at with
+ainΛ_0,
[
- v_sv_h=(omega(s,h))/(omega(a,t))v_av_t,qquad
+ v_sv_h=(omega(s,h))/(omega(a,t))v_av_t,
v_hv_t^*=(omega(s,h))/(omega(a,t))v_s^*v_a.
]
The other endpoint action commutes with every operation, so the two maps
@@ -1164,42 +1163,42 @@ compose to a bimodular isometry into two-sided induction, whose range is a
closed summand. The finite basis formulas show preservation of finite
module dimensions.
-If an element of $Gamma_0$ has finite conjugacy class in $Gamma_0$, its
-class in $Gamma$ is a finite union of translates of that class; ICC of
-$Gamma$ forces it to be the identity.
+If an element of Gamma_0 has finite conjugacy class in Gamma_0, its
+class in Gamma is a finite union of translates of that class; ICC of
+Gamma forces it to be the identity.
-For a source in $mathscr K$,scrK$, choose the actual isomorphism of finite-index
+For a source in mathscr K,scrK, choose the actual isomorphism of finite-index
subgroups of Definition~def:class; its image is a finite-index
subgroup of a reference lattice, hence itself a lattice in the same
ambient group. Pull the scalar cocycle through that isomorphism, without
-modifying it by a finite kernel. The restrictions retain property $(T)$
-and ICC, as well as bifiniteness; property $(T)$ permanence is used in its
+modifying it by a finite kernel. The restrictions retain property (T)
+and ICC, as well as bifiniteness; property (T) permanence is used in its
finite-covolume and finite-index forms cite[Theorem~1.7.1]{BHV2008}; see
also cite[Theorem~3]{DelarocheKirillov1968}. Lemmas~lem:setup-compact-reduction
and~lem:setup-finite-index-induction now reduce the geometric work
-to the actual-lattice, weakly mixing setting. Restoring $V_c$ after a
-partition of $K'_c$ tensors every atom with $V_c$ and restores the
+to the actual-lattice, weakly mixing setting. Restoring V_c after a
+partition of K'_c tensors every atom with V_c and restores the
original exact right multiplier, changing neither the atom permutations
nor their freeness; this is used only after the partition is proved.
## Tensor copies and complementary regular realizations
-From now until Section~sec:recovery, let $Gamma<G$ be an actual
+From now until Section~sec:recovery, let Gamma<G be an actual
ICC lattice satisfying Equation~(eq:setup-weak-mixing). Comparing two
-source copies uses two copies of $K$; comparing two comparisons sharing
+source copies uses two copies of K; comparing two comparisons sharing
their first copy uses three. The right comparison group acts diagonally.
We omit one left source action from the regular model so that its induced
-transports remain available on $G/Gamma$.
+transports remain available on G/Gamma.
-Take $m=1$ or $2$ and put
+Take m=1 or 2 and put
[
- H=K^{ ⊗ (m+1)},qquad
+ H=K^{ ⊗ (m+1)},
R=Gamma_1times...timesGamma_mtimesΛ.
]
-The slots of $H$ are numbered $0,...,m$; $pi_j$ denotes the left
-representation in slot $j$. Let $W$ be the representation of $R$ given by
-the $pi_j$, $jge1$, and diagonal $rho$, with split multiplier
+The slots of H are numbered 0,...,m; pi_j denotes the left
+representation in slot j. Let W be the representation of R given by
+the pi_j, jge1, and diagonal rho, with split multiplier
alpha_R((g_1,...,g_m,h),(g'_1,...,g'_m,h'))
=prod_{j=1}^mmu(g_j,g'_j)omega(h,h')bar^{,m+1}.
@@ -1208,85 +1207,85 @@ the $pi_j$, $jge1$, and diagonal $rho$, with split multiplier
[Complementary regularity]
-The representation $W$ is normal and finite over the twisted factor for
-$R$. The same holds with any one slot omitted instead of slot zero. The
-joint commutant of all $pi_j(Gamma)$, $0le jle m$, and diagonal
-$rho(Λ)$ is scalar.
+The representation W is normal and finite over the twisted factor for
+R. The same holds with any one slot omitted instead of slot zero. The
+joint commutant of all pi_j(Gamma), 0le jle m, and diagonal
+rho(Λ) is scalar.
Proof.
Embed the omitted slot in a finite regular multiple for its
-$overlineomega$-projective right representation. At that regular index
+overlineomega-projective right representation. At that regular index
absorb the diagonal right actions of the remaining slots by
Lemma~lem:setup-twisted-absorption. The absorbing unitaries are
-formed from the other slots' $rho(h)$ and commute with all their
-$pi_j(g)$. After absorption the $Λ$-action is regular with
-multiplier $overlineomega^{,m+1}$, and each remaining $pi_j$ embeds
-independently in a finite regular multiple for $L_mu(Gamma_j)$. The
+formed from the other slots' rho(h) and commute with all their
+pi_j(g). After absorption the Λ-action is regular with
+multiplier overlineomega^{,m+1}, and each remaining pi_j embeds
+independently in a finite regular multiple for L_mu(Gamma_j). The
result is a subrepresentation of a finite regular multiple for exactly
Equation~(eq:setup-tensor-multiplier), symmetrically in the omitted
slot.
-Write $P=pi(M)'$. The final joint commutant is
-$(P^{barbar ⊗⊗ (m+1)})^{Adrho^{ ⊗ (m+1)}}$, which
+Write P=pi(M)'. The final joint commutant is
+(P^{barbar ⊗⊗ (m+1)})^{Adrho^{ ⊗ (m+1)}}, which
Equation~(eq:setup-weak-mixing) makes scalar. For completeness: an
-invariant tensor with a factor in $L^2(P)ominusmathbb C1$bbC1$ gives a
+invariant tensor with a factor in L^2(P)ominusmathbb C1bbC1 gives a
Hilbert--Schmidt intertwiner between that factor's representation and
another unitary representation; composing with its adjoint gives a
positive compact operator with a nonzero finite-dimensional invariant
spectral space, contrary to weak mixing. Decomposing each tensor factor
-as $mathbb C1bbC1 ⊕ (L^2(P)ominusmathbb C1)$bbC1)$ applies this to every
+as mathbb C1bbC1 ⊕ (L^2(P)ominusmathbb C1)bbC1) applies this to every
nonconstant summand.
Choose a column realization
[
- H=eL^2(B)^k,qquad ein M_k(B),qquad
+ H=eL^2(B)^k, ein M_k(B),
d=(Tr_k ⊗ tau_B)(e)>0.
]
-Here $B$ is the opposite of the $alpha_R$-twisted factor of $R$; using
+Here B is the opposite of the alpha_R-twisted factor of R; using
the adjoints of the opposite canonical generators identifies it with the
-$alpha_Rbar$-twisted group factor, whose generators we write as
-$w_r$. Thus $W_r$ is right multiplication by $w_r^*$, which has
-multiplier $alpha_R$. Set
+alpha_Rbar-twisted group factor, whose generators we write as
+w_r. Thus W_r is right multiplication by w_r^*, which has
+multiplier alpha_R. Set
- Y=G/Gamma,qquad
- mathcal H=L^2(Y;H),calH=L^2(Y;H),qquad
+ Y=G/Gamma,
+ mathcal H=L^2(Y;H),calH=L^2(Y;H),
mathcal N=L^∞(Y)calN=L^∞(Y)overlineoverline ⊗⊗ eM_k(B)e.
-Give $Y$ its invariant probability measure. The trace on $mathcal N$calN$ is
-$tr_{mathcal N}calN(A)
-=d^{-1}int_Y(Tr_k ⊗ tau_B)(A(y)),dy$, whereas the Hilbert norm of
+Give Y its invariant probability measure. The trace on mathcal NcalN is
+tr_{mathcal N}calN(A)
+=d^{-1}int_Y(Tr_k ⊗ tau_B)(A(y)),dy, whereas the Hilbert norm of
a column uses the unnormalized entrywise trace sum. All finite-matrix
estimates permit rectangular products. Whenever a family of coefficient
algebras is varied, coefficient traces are normalized and matrix sizes
are fixed, or the Hilbert-norm factors are retained explicitly; an
operator-norm bound alone is not a bound independent of trace rescaling.
-Choose a Borel section $d_0:Yto G$ and write
+Choose a Borel section d_0:Yto G and write
[
- c(a,y)=d_0(ay)^{-1}ad_0(y),qquad
- X_a(y)=pi_0(c(a,y)),qquad
+ c(a,y)=d_0(ay)^{-1}ad_0(y),
+ X_a(y)=pi_0(c(a,y)),
(U_af)(ay)=X_a(y)f(y).
]
These transports compose up to a scalar function of modulus one on the
base. More explicitly,
-$X_a(by)X_b(y)=mu(c(a,by),c(b,y))X_{ab}(y)$. Conjugation on
+X_a(by)X_b(y)=mu(c(a,by),c(b,y))X_{ab}(y). Conjugation on
base-decomposable operators is therefore a genuine action, denoted
-$sigma=Ad U$.
+sigma=Ad U.
[Induced finite algebras]
-The transports $U_a$ are strongly continuous in $a$, and $sigma$
-preserves the trace on $mathcal N$.calN$. For $1le jle m$, let $R'_j$ be $R$
-with $Gamma_j$ omitted, put $beta_j=Ad W_{Gamma_j}$, and let
-$mathcal A_j$calA_j$ be the commutant of the base multipliers and
-$W|_{R'_j}$. Then
+The transports U_a are strongly continuous in a, and sigma
+preserves the trace on mathcal N.calN. For 1le jle m, let R'_j be R
+with Gamma_j omitted, put beta_j=Ad W_{Gamma_j}, and let
+mathcal A_jcalA_j be the commutant of the base multipliers and
+W|_{R'_j}. Then
[
- (mathcal A_j)^calA_j)^{beta_j}=mathcal N,calN,qquad
- (mathcal A_j)^calA_j)^sigma has a faithful finite $beta_j$-invariant
- trace,qquad
+ (mathcal A_j)^calA_j)^{beta_j}=mathcal N,calN,
+ (mathcal A_j)^calA_j)^sigma has a faithful finite beta_j-invariant
+ trace,
mathcal N^G=calN^G=mathbb C1.bbC1.
]
@@ -1294,26 +1293,26 @@ $W|_{R'_j}$. Then
Proof.
Translation is continuous in measure on measurable maps on the
finite-measure quotient; applied to the Borel section this shows
-$c(a,y)$ varies continuously in measure as a discrete-valued map. Verify
+c(a,y) varies continuously in measure as a discrete-valued map. Verify
first on compact sets where the section is continuous, by Lusin
approximation, then remove sets of small measure. The corresponding
projective unitaries are bounded, so their multiplication operators vary
-strongly on $mathcal H$.calH$. Conjugation of the corner is inner at each fibre
-and translation preserves base probability, so $sigma$ preserves the
-trace. The first identity is the commutant of all of $W$ in the
+strongly on mathcal H.calH. Conjugation of the corner is inner at each fibre
+and translation preserves base probability, so sigma preserves the
+trace. The first identity is the commutant of all of W in the
decomposable algebra.
-For the second, undo induction. If $A(y)$ is $sigma$-invariant, lift it,
-for $g=d_0(y)gamma$, as $widetilde A(g)=pi_0(gamma)^*A(y)pi_0(gamma)$.
-The cocycle covariance makes this field left $G$-invariant, hence almost
-everywhere constant with value $T$, and right $Gamma$-covariance says $T$
-commutes with $pi_0(Gamma)$ and with $W|_{R'_j}$. Conversely every such
-$T$ gives a fixed field. Thus $(mathcal A_j)^calA_j)^sigma$ is the commutant of
-the complementary regular representation obtained by omitting slot $j$;
+For the second, undo induction. If A(y) is sigma-invariant, lift it,
+for g=d_0(y)gamma, as widetilde A(g)=pi_0(gamma)^*A(y)pi_0(gamma).
+The cocycle covariance makes this field left G-invariant, hence almost
+everywhere constant with value T, and right Gamma-covariance says T
+commutes with pi_0(Gamma) and with W|_{R'_j}. Conversely every such
+T gives a fixed field. Thus (mathcal A_j)^calA_j)^sigma is the commutant of
+the complementary regular representation obtained by omitting slot j;
Lemma~lem:setup-normal-tensors makes it finite, with normalized
-$beta_j$-invariant trace since the remaining slot acts by unitaries in
+beta_j-invariant trace since the remaining slot acts by unitaries in
that commutant. Including all slots and using the last assertion of that
-lemma gives $mathcal N^G=calN^G=mathbb C1$.bbC1$.
+lemma gives mathcal N^G=calN^G=mathbb C1.bbC1.
## The source groups and full versus partial mixing
@@ -1328,14 +1327,14 @@ root-generated simplicity, and highest-weight theory
[Factors and panel groups]
At a complex place the local group is regarded as a real Lie group. The
-noncompact property-$(T)$ archimedean factors are the higher-rank ones and
-the groups locally isomorphic to $Sp(n,1)$, $nge2$, or $F_{4(-20)}$. At
+noncompact property-(T) archimedean factors are the higher-rank ones and
+the groups locally isomorphic to Sp(n,1), nge2, or F_{4(-20)}. At
a finite place the factors have relative rank at least two. For
-higher-rank property $(T)$ see cite[Theorem~6]{DelarocheKirillov1968};
+higher-rank property (T) see cite[Theorem~6]{DelarocheKirillov1968};
the real rank-one classification is in cite[Section~3.3 and
Theorem~3.5.4]{BHV2008}. Relative rank one at a finite place gives an
unbounded action on the Bruhat--Tits tree and is incompatible with
-property $(T)$ cite[Remark~1.6.3]{BHV2008}.
+property (T) cite[Remark~1.6.3]{BHV2008}.
In a rank-one semisimple Levi position exactly one almost simple factor is
isotropic. Its minimal boundary is the full panel residue. The compact and
@@ -1353,18 +1352,18 @@ or topological covering flag.
For the cover assertion, a central covering induces an isomorphism on the
corresponding relative root groups, so its image contains the subgroup
generated by the unipotent radicals cite[Theorem~2.20(iii) and
-Proposition~2.24(ii)]{BT1972Complements}. For a central kernel $C$, the
-rational-point quotient injects into $H^1(k,C)$, which is finite over the
+Proposition~2.24(ii)]{BT1972Complements}. For a central kernel C, the
+rational-point quotient injects into H^1(k,C), which is finite over the
present local fields cite[Propositions~1.12 and~1.17, Theorem~6.1]{BorelSerre1964}.
This proves the finite-index assertion directly.
[Product mixing and partial escape]
-For the action on $L^2(mathcal N)$,calN)$, escape in every simple factor gives
+For the action on L^2(mathcal N),calN), escape in every simple factor gives
weak operator convergence to the projection onto constants. If a path is
-the identity outside a subproduct $G_0$ and escapes in every factor of
-$G_0$, its weak limit is the projection onto $L^2(mathcal N)^calN)^{G_0}$,
-equivalently the conditional expectation onto $mathcal N^calN^{G_0}$. On the
+the identity outside a subproduct G_0 and escapes in every factor of
+G_0, its weak limit is the projection onto L^2(mathcal N)^calN)^{G_0},
+equivalently the conditional expectation onto mathcal N^calN^{G_0}. On the
orthogonal complement of a simple factor's invariants, its
matrix-coefficient decay remains uniform after composition with arbitrary
commuting group unitaries.
@@ -1376,13 +1375,13 @@ groups and their root-generated lifts from
Convention~conv:setup-panels [cite]; an independent
proof in this convention is cite[Theorem~1.1 and Definition~4.23]{Ciobotaru2015}.
We justify the type-I input for these actual groups. For an isotropic
-simple factor $mathbf H$bfH$ over $k$, choose a nonzero parabolic unipotent
-radical $mathbf U$.bfU$. The $k$-span of the $mathbf H(k)$-conjugatesbfH(k)$-conjugates of
-$Lie(mathbf U)$bfU)$ is a nonzero invariant Lie ideal, hence all
-of $Lie(mathbf H)$.bfH)$. Choose $X_1,...,X_d$ from these
+simple factor mathbf HbfH over k, choose a nonzero parabolic unipotent
+radical mathbf U.bfU. The k-span of the mathbf H(k)-conjugatesbfH(k)-conjugates of
+Lie(mathbf U)bfU) is a nonzero invariant Lie ideal, hence all
+of Lie(mathbf H).bfH). Choose X_1,...,X_d from these
conjugate subspaces; the map
-$(t_1,...,t_d)mapstoprod_jexp(t_jX_j)$ has invertible differential
-at zero and lands in $mathbf H(k)^+$.bfH(k)^+$. The local-field inverse function
+(t_1,...,t_d)mapstoprod_jexp(t_jX_j) has invertible differential
+at zero and lands in mathbf H(k)^+.bfH(k)^+. The local-field inverse function
theorem makes this subgroup open; at finite places see
cite[Proposition~1.11]{Gloeckner2008}. The same holds for the root-generated
subgroup of an algebraic simply connected cover. At an archimedean place
@@ -1393,18 +1392,18 @@ reductive group of local points is type~I [cite], and type~I
passes to its open root-generated subgroup
cite[Proposition~6.E.21(1)]{BekkaHarpe2019}.
-Restrict a separable unitary representation $pi$ to one simple factor $S$.
+Restrict a separable unitary representation pi to one simple factor S.
Its type-I decomposition with multiplicity is
[
mathcal H=calH=int_{widehatwidehat S}S^{ ⊕ }(H_xi ⊗ M_xi),deta(xi),
- qquad
+
pi(s)=int_{widehatwidehat S}S^{ ⊕ }(xi(s) ⊗ 1),deta(xi).
]
-In this central decomposition every unitary commuting with $S$ acts
-fibrewise as $1 ⊗ V_xi$
+In this central decomposition every unitary commuting with S acts
+fibrewise as 1 ⊗ V_xi
cite[Propositions~5.4.7 and~8.6.4, Theorem~8.6.6]{Dixmier1977}; see also
cite[Theorem~7.32]{Folland2015}. For finite tensor sums in a nontrivial
-fibre, uniformly over the $V_xi$,
+fibre, uniformly over the V_xi,
[
|bigl<(xi(s) ⊗ V_xi)sum_j e_j ⊗ u_j,
sum_k f_k ⊗ v_kbigr>|
@@ -1413,17 +1412,17 @@ fibre, uniformly over the $V_xi$,
Howe--Moore makes the right side tend to zero along escape; approximation
by finite tensor sums is uniform because all operators have norm one.
Dominated convergence then proves the uniform decay off the
-$S$-invariants.
+S-invariants.
-Let $P_i$ project onto the invariants of the $i$th simple factor. These
+Let P_i project onto the invariants of the ith simple factor. These
commute with every group action and each other. Expand the identity as the
finite sum of orthogonal products obtained from
-$prod_i(P_i+(1-P_i))$. In every sector containing $1-P_i$, the uniform
+prod_i(P_i+(1-P_i)). In every sector containing 1-P_i, the uniform
decay kills its coefficients when that factor escapes. Thus all-factor
-escape gives the weak limit $prod_iP_i$, the projection onto constants
+escape gives the weak limit prod_iP_i, the projection onto constants
by Proposition~prop:setup-induced-algebras. For partial escape,
-expand only over the factors of $G_0$; the limit is the projection onto
-$L^2(mathcal N)^calN)^{G_0}$, i.e. the trace-preserving conditional
+expand only over the factors of G_0; the limit is the projection onto
+L^2(mathcal N)^calN)^{G_0}, i.e. the trace-preserving conditional
expectation.
@@ -1434,41 +1433,41 @@ on a Cartan path exact root coordinates will determine which flag
coordinates survive.
An archimedean panel inside a higher-rank component cannot have noncompact
-Lie algebra $mathfrak f_frakf_{4(-20)}$: its complexification would have to
+Lie algebra mathfrak f_frakf_{4(-20)}: its complexification would have to
occur as a component of a proper absolute Dynkin subdiagram, and the
Dynkin list has no such occurrence. At a finite place the effective
-rank-one Lie algebra is simple also over $mathbb Q_p$:bbQ_p$: after restriction
+rank-one Lie algebra is simple also over mathbb Q_p:bbQ_p: after restriction
of scalars its absolutely simple factors form one Galois orbit, so a
-$mathbb Q_p$-idealbbQ_p$-ideal is either zero or the whole algebra. An adjoint
+mathbb Q_p-idealbbQ_p-ideal is either zero or the whole algebra. An adjoint
automorphism centralizing an open subgroup centralizes all sufficiently
small inner exponentials, hence this Lie algebra, and is the identity.
These assertions concern the ambient local groups, not a discreteness
assumption on a lattice projection.
-Let $Pi$ be the union of the simple indivisible relative roots of
-the factors. A face of type $IsubsetPi$ has strictly positive simple
-coordinates exactly in $I$; opposition of types is denoted by a star.
+Let Pi be the union of the simple indivisible relative roots of
+the factors. A face of type IsubsetPi has strictly positive simple
+coordinates exactly in I; opposition of types is denoted by a star.
Choose a special maximal compact in each factor. Cartan and Iwasawa
decompositions can be made on the simply connected covers and descended.
-Write $mu(g)$ for the Cartan vector only when its argument is a
+Write mu(g) for the Cartan vector only when its argument is a
local-group element; this is distinct from the scalar two-cocycle
-$mu(g,h)$. For a positive root $alpha$ put
+mu(g,h). For a positive root alpha put
[
- m_alpha=dim_{k}mathfrak g_frakg_alpha at a finite place,qquad
+ m_alpha=dim_{k}mathfrak g_frakg_alpha at a finite place,
2rho=sum_{alpha>0}m_alphaalpha,
]
and use the analogous real restricted-root multiplicities at archimedean
-places. Double roots are included. The fundamental coweight rays $e_i$
+places. Double roots are included. The fundamental coweight rays e_i
are dual to the simple indivisible roots.
[Exact finite-place translation coordinates]
-At a finite place use the module absolute value with $|varpi|=q^{-1}$.
+At a finite place use the module absolute value with |varpi|=q^{-1}.
The Cartan vector is the actual dominant translation vector of a Cartan
representative in the split centralizer: its pairing with every
centralizer character is the exact logarithm of that character's module.
We do not replace this by a coarsely equivalent length in identities
-involving heights. All displacements and offsets along $e_i$ range on a
+involving heights. All displacements and offsets along e_i range on a
sufficiently divisible mesh in this logarithmic translation lattice.
Cocharacter rationality and common multiples of relative coroot
translations and central isogenies ensure these elements and their
@@ -1482,30 +1481,30 @@ split displacement.
[Haar size of Cartan shells]
-For a bounded-thickness shell about a dominant vector $L$, its Haar
-measure is at most $Cexp(2rho(L))$. For one fixed sufficiently large
-thickness there is a lower bound $cexp(2rho(L))$. The constants are
-independent of $L$ on the permissible Cartan mesh. The analogous
-assertions hold for the finite product, with the sum of the factors' $2rho$.
+For a bounded-thickness shell about a dominant vector L, its Haar
+measure is at most Cexp(2rho(L)). For one fixed sufficiently large
+thickness there is a lower bound cexp(2rho(L)). The constants are
+independent of L on the permissible Cartan mesh. The analogous
+assertions hold for the finite product, with the sum of the factors' 2rho.
Proof.
At a real place use the Cartan density, a product of the appropriate
-powers of $sinhalpha(L)$. The upper bound follows on every fixed bounded
-neighborhood. For the lower bound, even when $L$ lies on a wall, choose
+powers of sinhalpha(L). The upper bound follows on every fixed bounded
+neighborhood. For the lower bound, even when L lies on a wall, choose
inside the shell one fixed bounded offset box whose simple coordinates are
bounded away from zero; on this box every positive-root factor is bounded
below by a fixed multiple of its exponential, and the box has fixed
positive Euclidean volume.
At a finite place the Cartan double coset has measure
-$[K:Kcap aKa^{-1}]$ times the fixed compact measure, comparable to
-$exp(2rho(L))$. In compact root charts a sufficiently small analytic
+[K:Kcap aKa^{-1}] times the fixed compact measure, comparable to
+exp(2rho(L)). In compact root charts a sufficiently small analytic
compact open subgroup has product coordinates in negative-root,
centralizer, and positive-root directions; under conjugation by the
dominant translation its intersection shrinks the expanding root
lattices, and the index is the product of their module indices with
-exponent $sum_{alpha>0}m_alphaalpha(L)$. Equivariant exponentials and
+exponent sum_{alpha>0}m_alphaalpha(L). Equivariant exponentials and
parabolic product uniqueness give this calculation; changing the compact
frame changes the small subgroups by a uniformly bounded index. Other
Cartan representatives are within bounded distance of the chosen split
@@ -1527,7 +1526,7 @@ corresponding relative simple root; every other weight drops by a
nonnegative combination involving that root. Consequently a normalized
Cartan matrix has rank-one limit exactly when that simple gap diverges.
-At a finite place these are $k$-rational modules, used only to form compact
+At a finite place these are k-rational modules, used only to form compact
projective matrix data: use compact-invariant lattice norms and compact
sup-norm frames, recording both the matrix and its inverse up to projective
normalization. At an archimedean target, for Hilbert highest-line
@@ -1538,9 +1537,9 @@ field.
[Face blocks and residual projectivities]
-Let a sequence have diverging simple gaps of types $D$ and keep its
+Let a sequence have diverging simple gaps of types D and keep its
complementary gaps in a bounded window. Its compactified matrix data
-determine an image $D$-flag, a reverse $D^*$-flag, and an invertible
+determine an image D-flag, a reverse D^*-flag, and an invertible
residual Levi transport between their transverse refinement spaces,
independent of compact frame choices. On a deeper stratum the already
determined image vertices persist. On compact transverse windows the
@@ -1591,19 +1590,19 @@ denominators are bounded away from zero on a buffered larger window.
[Interpolation in an affine building]
-Fix an endpoint $x$ in one of the affine buildings above. In an apartment
-containing $x,y$, multiply the dominant simple coordinates of $y-x$ by
-independent numbers $t_iin[0,1]$. The resulting point is independent of
-the apartment and depends on $y$ with a uniform Lipschitz constant. The
-same holds for variation of $x$, after opposition and replacement of the
+Fix an endpoint x in one of the affine buildings above. In an apartment
+containing x,y, multiply the dominant simple coordinates of y-x by
+independent numbers t_iin[0,1]. The resulting point is independent of
+the apartment and depends on y with a uniform Lipschitz constant. The
+same holds for variation of x, after opposition and replacement of the
fractions by their complements. Constants for a finite product depend only
on its root systems.
Proof.
-Write $L_i=alpha_i(y-x)ge0$ and let $T_t$ have simple coordinates
-$t_iL_i$. For every positive root $alpha=sum_in_ialpha_i$, where
-$n_ige0$,
+Write L_i=alpha_i(y-x)ge0 and let T_t have simple coordinates
+t_iL_i. For every positive root alpha=sum_in_ialpha_i, where
+n_ige0,
[
0lealpha(T_t(y-x))=sum_in_it_iL_ilesum_in_iL_i=alpha(y-x).
]
@@ -1611,7 +1610,7 @@ Thus the new point lies in every affine root half-apartment containing both
endpoints, i.e. in their root enclosure. Apartment intersections are
enclosed; the transition maps agree on this enclosure, and the Weyl
stabilizer of a wall displacement fixes its transformed displacement. This
-proves independence. On any apartment and Weyl sector $T_t$ has uniformly
+proves independence. On any apartment and Weyl sector T_t has uniformly
bounded Euclidean operator norm; subdivide a segment of varying endpoints
into finitely many cells and Weyl sectors, sum the bounds, and use
apartment independence at the interfaces. Reversing the endpoints applies
@@ -1630,14 +1629,14 @@ any action of the comparison group on a building.
[Operators and Fourier coordinates]
-An invariance under $U$ is a conjugation invariance of decomposable
+An invariance under U is a conjugation invariance of decomposable
operators, so base scalar phases do not change it. An ``other group'' in a
-commutant includes $Λ$ unless $Λ$ is explicitly the measured
-group. A target Fourier arrow records $w_{(h,cdot)}$ and the Cartan or
-flag data of $hinGamma$ in a measured source-lattice copy; it does not
-record geometric data of the $Λ$ index. A bounded column is an
+commutant includes Λ unless Λ is explicitly the measured
+group. A target Fourier arrow records w_{(h,cdot)} and the Cartan or
+flag data of hinGamma in a measured source-lattice copy; it does not
+record geometric data of the Λ index. A bounded column is an
operator-bounded tracial matrix column, essentially uniformly bounded when
-it depends on $y$. The closed range of a positive effect means its support
+it depends on y. The closed range of a positive effect means its support
space when a range inclusion is asserted.
@@ -1651,26 +1650,26 @@ these measurements and their input forms, and to prove that a vertex
translation produces exactly one divergent target gap while successively
completing such a vertex-origin path adds exactly one target gap per stage.
-We retain the regular realizations of Section~sec:setup: $H=eL^2(B)^k$,
-$mathcal H=L^2(Y;H)$,calH=L^2(Y;H)$, $d=(Tr ⊗ tau_B)(e)$. For an integrable corner
+We retain the regular realizations of Section~sec:setup: H=eL^2(B)^k,
+mathcal H=L^2(Y;H),calH=L^2(Y;H), d=(Tr ⊗ tau_B)(e). For an integrable corner
field write
[
- mathsf T(C)=sfT(C)=int_Y(Tr ⊗ tau_B)(C(y)),dy,qquad
+ mathsf T(C)=sfT(C)=int_Y(Tr ⊗ tau_B)(C(y)),dy,
tau_{mathcal N}calN(C)=d^{-1}mathsf T(C).sfT(C).
]
-Thus $tau_{mathcal N}calN$ is a probability trace while the Hilbert norm of a
-column is $|f|_2^2=mathsf T(ff^*)$.sfT(ff^*)$. Matrix sizes in this section are
+Thus tau_{mathcal N}calN is a probability trace while the Hilbert norm of a
+column is |f|_2^2=mathsf T(ff^*).sfT(ff^*). Matrix sizes in this section are
fixed and finite. Operator-bounded columns form a dense subspace. The
constant corner frame is
[
- b_r=eε_r,qquad 1le rle k,qquad
+ b_r=eε_r, 1le rle k,
sum_{r=1}^k b_rb_r^*=e.
]
## Compact matrix data and the surviving flags
-Write $Pi$ for the indivisible relative simple roots. A face of type
-$IsubsetPi$ has strictly positive simple coordinates exactly in $I$;
+Write Pi for the indivisible relative simple roots. A face of type
+IsubsetPi has strictly positive simple coordinates exactly in I;
opposition of types is denoted by a star. Partial flags are transverse when
they are in the open relative position of subfaces of opposite chambers;
incidence means containment in one chamber.
@@ -1680,20 +1679,20 @@ incidence means containment in one chamber.
There is a finite family of projectively normalized matrix coordinates,
including coordinates of the inverse, with the following properties.
[label=(roman*)]
-item The set $DsubsetPi$ of diverging simple gaps is detected by rank one
-in the fundamental highest-line coordinates. A stratum with gap set $D$
-carries an image $D$-flag, a reverse $D^*$-flag, and a residual projectivity
+ The set DsubsetPi of diverging simple gaps is detected by rank one
+in the fundamental highest-line coordinates. A stratum with gap set D
+carries an image D-flag, a reverse D^*-flag, and a residual projectivity
between their Levi refinement spaces. Its closure has only gap sets
-containing $D$.
-item On a bounded complementary-gap window the residual blocks are
+containing D.
+ On a bounded complementary-gap window the residual blocks are
invertible in compact frames. They send a full flag transverse to the
reverse face, by transverse parabolic projection followed by Levi transport,
to a full flag incident with the image face.
-item Fixed left and right group multiplications preserve gap sets. Right
+ Fixed left and right group multiplications preserve gap sets. Right
multiplication leaves the image flag unchanged; left multiplication
transports it by the usual target action. Reverse data obey the opposite
rule, and residual projectivities change by the corresponding compositions.
-item In an archimedean highest-line module each surviving top face space is
+ In an archimedean highest-line module each surviving top face space is
spanned by the highest lines of flags incident with the observed face. A
rank-one Levi pencil has the effective rank-one projectivity group and its
omitted simple gap, up to the bounded changes made by compact endpoint
@@ -1715,10 +1714,10 @@ divergence of the corresponding simple gap; the finitely many coordinates
record the gap set, its image and reverse flags, and residual maps. If an
additional gap diverges the previous image vertices persist by that lemma.
-For the multiplication rule, both $a$ and $a^{-1}$ have finite norm; thus
-projectively normalized multiplication by $a$ has denominator bounded away
+For the multiplication rule, both a and a^{-1} have finite norm; thus
+projectively normalized multiplication by a has denominator bounded away
from zero, so fixed multiplication changes each simple Cartan gap by a
-bounded amount. Left multiplication carries the image top space by $a$;
+bounded amount. Left multiplication carries the image top space by a;
right multiplication leaves it unchanged. The same calculation for inverse
matrices gives the reverse-flag rule. The residual map transforms by the
corresponding compositions.
@@ -1747,8 +1746,8 @@ corner projection before the next operation. This permits a diagonal
spectral calculus without claiming that an angular Fourier cutoff preserves
the operator norm of a column.
-Choose a path $a$ with $a^{-1}$ split toward a source face $F$ of type $I$,
-its coordinates in $I$ tending to infinity. A lexicographic face path
+Choose a path a with a^{-1} split toward a source face F of type I,
+its coordinates in I tending to infinity. A lexicographic face path
starts with a vertex translation and adds one new simple coordinate at each
successively slower scale; the fastest parameter is taken to its limit first
while slower parameters stay fixed, then the next limit, and so on. A
@@ -1757,8 +1756,8 @@ the gap upper bound below applies to both, though the exact successive count
is asserted for vertex-origin chains. Finite-place parameters live on the
divisible cocharacter meshes of Section~sec:setup.
-The measured label is the compact matrix data of $h^{-1}$, where $h$ is the
-Fourier index in the chosen copy $Gamma_j$. Other group indices remain
+The measured label is the compact matrix data of h^{-1}, where h is the
+Fourier index in the chosen copy Gamma_j. Other group indices remain
coefficient indices. On full regular arrays, a scalar test of this label is
a diagonal Hilbert multiplier; these tests are not followed by corner
compression before another array operation.
@@ -1768,12 +1767,12 @@ The one-column map
(V_a f)(y)=X_a(y)f(y)
]
is an isometry with the integrated tracial Hilbert norm, so its Hilbert
-ultralimit $V$ is an isometry. On the minimal spectral span of $Vmathcal H$calH$
-the continuous diagonal tests give a PVM $E^{(1)}$; the compression
+ultralimit V is an isometry. On the minimal spectral span of Vmathcal HcalH
+the continuous diagonal tests give a PVM E^{(1)}; the compression
[
Phi(A)=V^*E^{(1)}(A)V
]
-is a positive effect on the original input. For $A$ recording only a gap
+is a positive effect on the original input. For A recording only a gap
stratum and its image flag this is the image measurement; it need not be a
projection.
@@ -1783,7 +1782,7 @@ The two-column map, initially on operator-bounded columns, is
=sqrt d,(b(ay)^*X_a(y)f(y))^{lim}.
-Its output has its own diagonal spectral measure $E$. The link identities
+Its output has its own diagonal spectral measure E. The link identities
below determine the norm of this map and the Hilbert inputs to which it
extends. Iterated ultraproducts realize the specified successive limits;
restrict to the minimal closed spectral span of the input ranges and a
@@ -1811,26 +1810,26 @@ test therefore intertwines the shift in the ultralimit, and equality on
continuous functions extends to Borel spectral measures. Base multiplication
and the reindexing commute with diagonal tests.
-If a Fourier index $h$ is changed to $rh$ by a left multiplier, the recorded
-inverse changes to $h^{-1}r^{-1}$; the gap set and image flag are unchanged
-in the limit. Let $D$ be a bounded test of those data and $T$ a bounded left
+If a Fourier index h is changed to rh by a left multiplier, the recorded
+inverse changes to h^{-1}r^{-1}; the gap set and image flag are unchanged
+in the limit. Let D be a bounded test of those data and T a bounded left
Fourier multiplier with uniformly bounded finite Fourier approximants
-$T_lto T$ in trace $L^2$. In a form tested on uncut operator-bounded
-columns, replacing either occurrence of $T$ by $T_l$ before moving $D$
-costs at most $C|D|,|T-T_l|_2$, uniformly in the fixed column bounds
-and matrix sizes. For each finite Fourier sum, move $D$ through its
+T_lto T in trace L^2. In a form tested on uncut operator-bounded
+columns, replacing either occurrence of T by T_l before moving D
+costs at most C|D|,|T-T_l|_2, uniformly in the fixed column bounds
+and matrix sizes. For each finite Fourier sum, move D through its
individual shifts by the already proved limiting identity; the remaining
-product is the uncut Gram factor $T_l^*T_l$. Replacing this by $T^*T$ costs
+product is the uncut Gram factor T_l^*T_l. Replacing this by T^*T costs
another quantity tending to zero since
-$|T_l^*T_l-T^*T|_2le(|T_l|_∞+|T|_∞)|T_l-T|_2$. All errors
+|T_l^*T_l-T^*T|_2le(|T_l|_∞+|T|_∞)|T_l-T|_2. All errors
are estimated before applying the cutoff to the columns, so the argument
-applies to partial unitaries $T^*T=e$ and assigns no operator bound to $Df$.
+applies to partial unitaries T^*T=e and assigns no operator bound to Df.
-For a field reindexed at $ay$, invariant probability on $Y$ gives
-$int|T(ay)-T_l(ay)|_2^2dy=|T-T_l|_2^2$, so the same estimates hold after
+For a field reindexed at ay, invariant probability on Y gives
+int|T(ay)-T_l(ay)|_2^2dy=|T-T_l|_2^2, so the same estimates hold after
path transport. For a compact family of fixed translations, the cocycle
formula and continuity of translations in measure make their matrix fields
-compact in trace $L^2$; a finite net gives uniform finite-Fourier
+compact in trace L^2; a finite net gives uniform finite-Fourier
approximation of the compact family. No common finite-Fourier approximation
of the escaping paths themselves is used.
@@ -1840,7 +1839,7 @@ function of the base in each composition formula; pulling a fixed source
translation from one end of an array to the other therefore multiplies the
whole array by one scalar of modulus one, which cancels between the two
entries of a sesquilinear form. The same observation applies to a quotient
-path: after base reindexing, $X_tX_a^*$ is the transporter for $ta^{-1}$
+path: after base reindexing, X_tX_a^* is the transporter for ta^{-1}
times a common scalar, and its localized Fourier norm is unchanged. Products
in these identities are actual twisted matrix products; their coefficient
phases cannot be replaced by absolute values of untwisted convolutions.
@@ -1853,8 +1852,8 @@ without further commutation assumptions.
## Link forms, scalar masses, and the finite frame
-For a stratum of positive scalar mass $m_D$, divide its image measurement by
-$m_D$ to obtain a unital measurement. Scalarity is part of the next
+For a stratum of positive scalar mass m_D, divide its image measurement by
+m_D to obtain a unital measurement. Scalarity is part of the next
proposition, not an assumption on the construction.
[Link identities]
@@ -1862,19 +1861,19 @@ proposition, not an assumption on the construction.
The one-column measurements and the forms (eq:input-1) have the
following properties.
[label=(roman*)]
-item The image measurement in one measured copy lies in $mathcal A_j$,calA_j$,
-is $beta_j$-equivariant, and is invariant under the source face parabolic.
-Its gap-stratum masses belong to $mathcal N$calN$ and are scalar. Conjugating
-the source split frame transports the measurement by $sigma$.
-item If the path escapes in every simple component, the link is an isometry
-$mathcal HbarcalHbar ⊗ mathcal HcalHtomathcal K$.calK$. Its output forms are
+ The image measurement in one measured copy lies in mathcal A_j,calA_j,
+is beta_j-equivariant, and is invariant under the source face parabolic.
+Its gap-stratum masses belong to mathcal NcalN and are scalar. Conjugating
+the source split frame transports the measurement by sigma.
+ If the path escapes in every simple component, the link is an isometry
+mathcal HbarcalHbar ⊗ mathcal HcalHtomathcal K.calK. Its output forms are
decomposable at the two bases, covariant under the two independent target
shifts, invariant under the independently contracted source radicals, and
invariant under simultaneous source centralizers. An image-and-stratum
-effect depends only on the start input and is $1 ⊗ Phi(A)$; the reverse
+effect depends only on the start input and is 1 ⊗ Phi(A); the reverse
rule is symmetric.
-item If the path is active precisely in a subproduct $G_0$, put
-$N_0=mathcal N^calN^{G_0}$ and let $E_{N_0}:mathcal NcalNto N_0$ be the
+ If the path is active precisely in a subproduct G_0, put
+N_0=mathcal N^calN^{G_0} and let E_{N_0}:mathcal NcalNto N_0 be the
trace-preserving conditional expectation. On bounded columns the link has
|J(overlineoverline b,f)b,f)|^2
@@ -1891,7 +1890,7 @@ Its polarized extension on finite sums,
is a positive semidefinite form. The same Gram covariances hold on this form
space; they do not assert a tensor-product isometry. Hilbert extensions are
in Lemma~lem:links-relative-boundedness.
-item For any Borel event $A$ determined by a gap stratum and its image flag,
+ For any Borel event A determined by a gap stratum and its image flag,
the finite-frame identity is
< f,Phi(A)f>
@@ -1903,7 +1902,7 @@ additional limiting coordinates unchanged by the ignored left Fourier
shifts. When a proof starts from windows that also constrain the reverse
flag or residual projectivity, those windows must first be exhausted to such
an invariant event.
-item The image measurements are independent of the finite regular
+ The image measurements are independent of the finite regular
embedding. Adding another original tensor slot gives their normal copies,
tensored with the identity on that slot. This concerns the compressed
measurements and the operators determined by their stated one-ended
@@ -1916,74 +1915,74 @@ translate must tend to the identity in that order.
Proof.
(i) Base multipliers and companion actions commute with the relevant
diagonal tests and intertwine the input column map, so the compressed image
-effects lie in $mathcal A_j$.calA_j$. A measured target shift changes the inverse
+effects lie in mathcal A_j.calA_j. A measured target shift changes the inverse
Fourier label by the corresponding left geometric action, giving
-$beta_j$-equivariance via Lemma~lem:links-gram-transport.
+beta_j-equivariance via Lemma~lem:links-gram-transport.
-Let $n$ be in the radical contracted by the path. Sliding $U_n$ through
-$U_a$ puts $U_{ana^{-1}}$ at the far end up to a common base phase; this
+Let n be in the radical contracted by the path. Sliding U_n through
+U_a puts U_{ana^{-1}} at the far end up to a common base phase; this
tends strongly to the identity. Its discrete section cocycle is the identity
off a set of measure tending to zero, after its pure base shift is undone,
so on bounded inputs both descriptions give a vanishing Hilbert error even
with a bounded diagonal test inserted. This proves radical invariance of
the full limiting data.
-For a fixed centralizing $s$, the identity slides $U_s$ to the far end with
-the same $s$. Base reindexing has no effect on an integrated form, and the
+For a fixed centralizing s, the identity slides U_s to the far end with
+the same s. Base reindexing has no effect on an integrated form, and the
left corner multiplier is removed by Lemma~lem:links-gram-transport;
the uncut Gram factor is the corner identity. Thus the image measurement is
centralizer invariant. Radical and centralizer together generate the face
parabolic; conjugation of a split path removes the end conjugation.
-The mass of a gap stratum is target invariant, hence in $mathcal N$,calN$, and is
+The mass of a gap stratum is target invariant, hence in mathcal N,calN, and is
invariant under the entire face parabolic. Every inactive simple factor is
in the centralizer; in an active factor the contracted radical contains an
escaping sequence. By Howe--Moore applied to the representation on
-$L^2(mathcal N)$,calN)$, a radical-invariant vector orthogonal to the
+L^2(mathcal N),calN), a radical-invariant vector orthogonal to the
factor-invariant subspace would have a matrix coefficient both constant and
tending to zero. It is therefore factor invariant, and the mass is invariant
-under all simple factors, hence scalar because $mathcal N^G=calN^G=mathbb C1$.bbC1$.
+under all simple factors, hence scalar because mathcal N^G=calN^G=mathbb C1.bbC1.
This argument covers a pure vertex and a mixed path alike.
-(ii) For bounded $b,f$, cyclicity and the definition of the induced
+(ii) For bounded b,f, cyclicity and the definition of the induced
action give
[
|J(overlineoverline b,f)b,f)|^2
=dlim_amathsf T(ff^*sfT(ff^*sigma_{a^{-1}}(bb^*)).
]
On a mixed path, trace mixing makes the limit
-$d^2tau_{mathcal N}calN(ff^*)tau_{mathcal N}calN(bb^*)=|f|_2^2|b|_2^2$.
+d^2tau_{mathcal N}calN(ff^*)tau_{mathcal N}calN(bb^*)=|f|_2^2|b|_2^2.
Polarization gives
[
Bigl|sum_iJ(b_ibar,f_i)Bigr|^2
=d^2sum_{i,l}tau_{mathcal N}calN(f_lf_i^*)tau_{mathcal N}calN(b_ib_l^*),
]
-which is the norm on $mathcal HbarcalHbar ⊗ mathcal H$;calH$; the factor
-$sqrt d$ in (eq:input-1) is essential for this normalization.
+which is the norm on mathcal HbarcalHbar ⊗ mathcal H;calH; the factor
+sqrt d in (eq:input-1) is essential for this normalization.
Multiplication of an input by a start (resp. end) base function is
-represented on arrays by its value at $y$ (resp. $ay$); diagonal tests
+represented on arrays by its value at y (resp. ay); diagonal tests
commute with both, so their limiting forms are decomposable at the two
bases. Independent target shifts are handled by Fourier transport; source
radicals slide to contracted translates and cancel; simultaneous
centralizers use exact precomposition and finite-Fourier Gram. For an
image-and-stratum event, the ignored end Fourier shifts put each end slice
-in $mathcal Nbar$,calNbar$, and independent reverse-radical invariance makes
+in mathcal Nbar,calNbar, and independent reverse-radical invariance makes
it scalar by the Howe--Moore argument above; the compressed effect is
-$1 ⊗ C$, and the frame computation below identifies $C=Phi(A)$.
+1 ⊗ C, and the frame computation below identifies C=Phi(A).
-(iii) For an escape in $G_0$, the weak limit of the trace-preserving
-action on $L^2(mathcal N)$calN)$ is the projection onto $L^2(N_0)$, which yields
+(iii) For an escape in G_0, the weak limit of the trace-preserving
+action on L^2(mathcal N)calN) is the projection onto L^2(N_0), which yields
(eq:links-relative-form) and (eq:links-polarized-relative) as
limits of squared norms. The mass calculation above is separate; it does
-not replace $N_0$ by scalars.
+not replace N_0 by scalars.
(iv) For a continuous test of the image and stratum, approximate each
-$b_r(ay)^*$ by bounded finite Fourier matrices. The inverse-index convention
+b_r(ay)^* by bounded finite Fourier matrices. The inverse-index convention
makes the test invariant under their left Fourier shifts, so
Lemma~lem:links-gram-transport moves the test past the multipliers.
-Summing the uncut Gram factors gives $sum_rb_r(ay)b_r(ay)^*=e$; the
-two-column normalization contributes $d$, proving (eq:links-frame).
+Summing the uncut Gram factors gives sum_rb_r(ay)b_r(ay)^*=e; the
+two-column normalization contributes d, proving (eq:links-frame).
Equality of the limiting spectral measures extends it to invariant Borel
events. The proof uses neither factorization of the end variable nor mixed
trace mixing.
@@ -2004,11 +2003,11 @@ Lemma~lem:links-gram-transport identifies the two compressed
measurements. For the normal-copy assertion, begin with a regular-multiple
embedding for the first two slots and enlarge it by the entire extra slot.
Conjugating the extra slot by the projective inverse of its simultaneous
-$rho(h)$ action at the regular $Λ$-index transfers its multiplier to
-the regular $Λ$ multiplier via projective absorption, and this
-conjugation commutes with the extra $Gamma$-action, which can be embedded in
+rho(h) action at the regular Λ-index transfers its multiplier to
+the regular Λ multiplier via projective absorption, and this
+conjugation commutes with the extra Gamma-action, which can be embedded in
its own finite regular multiple. Both operations commute with scalar Fourier
-cutoffs on the original $Gamma_j$-coordinate. In original input coordinates
+cutoffs on the original Gamma_j-coordinate. In original input coordinates
the one-ended path acts only in slot zero, so its compressed tests are the
original tests tensored with identity on the added slot. The
embedding-independence identifies this with any subsequent realization.
@@ -2020,25 +2019,25 @@ all uncompressed spectral arrays.
## Relative boundedness and Hilbert inputs
The two norm formulas give different input spaces. On an all-factor path,
-$J$ extends to the ordinary Hilbert tensor product. On a partial path its
-norm still contains $E_{N_0}(ff^*)$, which can be unbounded for a Hilbert
+J extends to the ordinary Hilbert tensor product. On a partial path its
+norm still contains E_{N_0}(ff^*), which can be unbounded for a Hilbert
column. The useful replacement is to bound this conditional expectation for
one input.
[Relative boundedness and Hilbert inputs]
-For a path supported in $G_0$, call a column $f$ relatively bounded if the
-positive affiliated operator $E_{N_0}(ff^*)$ is bounded. Then:
+For a path supported in G_0, call a column f relatively bounded if the
+positive affiliated operator E_{N_0}(ff^*) is bounded. Then:
[label=(roman*)]
-item for fixed relatively bounded $f$, the link extends continuously to
-every Hilbert end column $b$, and conversely;
-item the image measurements of paths in factors of $G_0$ commute with
-$N_0$; their generated algebra, matrix effects, and support projections
+ for fixed relatively bounded f, the link extends continuously to
+every Hilbert end column b, and conversely;
+ the image measurements of paths in factors of G_0 commute with
+N_0; their generated algebra, matrix effects, and support projections
preserve relative boundedness;
-item a closed $N_0$-invariant subspace has a dense relatively bounded part,
+ a closed N_0-invariant subspace has a dense relatively bounded part,
and a relatively bounded column has operator-bounded approximants with the
same bound on its conditional expectation;
-item the link covariances hold on relatively bounded pairs and when one
+ the link covariances hold on relatively bounded pairs and when one
input is operator-bounded and the other a Hilbert column. A one-column gap
inclusion valid for all inputs also holds for bounded pairs in
(eq:input-1).
@@ -2046,38 +2045,38 @@ inclusion valid for all inputs also holds for bounded pairs in
Proof.
-Conditional expectation extends to positive $L^1$ elements. Put
-$A=E_{N_0}(ff^*)$; if $A$ is bounded, (eq:links-relative-form) gives
+Conditional expectation extends to positive L^1 elements. Put
+A=E_{N_0}(ff^*); if A is bounded, (eq:links-relative-form) gives
|J(overlineoverline b,f)b,f)|^2le d|A|_∞|b|_2^2.
-This proves the continuous extension from bounded $b$; the link limit is
+This proves the continuous extension from bounded b; the link limit is
always taken before this extension.
-Let $cin N_0$. Along a path in $G_0$, $c$ is a fixed source-invariant left
+Let cin N_0. Along a path in G_0, c is a fixed source-invariant left
multiplier; move it from the input to the end and remove it from an image
test by the finite Fourier--Gram argument, as a centralizer multiplier.
-Polarization shows every image effect commutes with $c$. If a bounded $Q$
-commutes with $N_0$, then for positive $cin N_0$,
-$< Qf,cQf>le|Q|^2< f,cf>$; the
+Polarization shows every image effect commutes with c. If a bounded Q
+commutes with N_0, then for positive cin N_0,
+< Qf,cQf>le|Q|^2< f,cf>; the
characterization of the conditional expectation by these tests implies
-$E_{N_0}((Qf)(Qf)^*)le|Q|^2E_{N_0}(ff^*)$. This holds on the generated
+E_{N_0}((Qf)(Qf)^*)le|Q|^2E_{N_0}(ff^*). This holds on the generated
algebra, its support projections, and after finite matrix amplification.
-For a closed subspace invariant under $N_0$, the spectral projections
-$p_n=1_{[0,n]}(E_{N_0}(ff^*))$ preserve it; $p_nf$ is relatively bounded and
-converges to $f$ since
-$|(1-p_n)f|_2^2=dtau_{mathcal N}calN((1-p_n)E_{N_0}(ff^*))to0$. For
-relatively bounded $f$, $f_n=1_{[0,n]}(ff^*)f$ gives operator-bounded
+For a closed subspace invariant under N_0, the spectral projections
+p_n=1_{[0,n]}(E_{N_0}(ff^*)) preserve it; p_nf is relatively bounded and
+converges to f since
+|(1-p_n)f|_2^2=dtau_{mathcal N}calN((1-p_n)E_{N_0}(ff^*))to0. For
+relatively bounded f, f_n=1_{[0,n]}(ff^*)f gives operator-bounded
columns converging in Hilbert norm with
-$E_{N_0}(f_nf_n^*)le E_{N_0}(ff^*)$.
+E_{N_0}(f_nf_n^*)le E_{N_0}(ff^*).
-These approximants converge in the link form against a fixed Hilbert $b$:
-first replace $b$ by bounded $b_l$; then (eq:links-relative-form)
-bounds the error by $d|b_l|_∞^2|f_n-f|_2^2$ on the tail while
-$E_{N_0}((f_n-f)(f_n-f)^*)le4E_{N_0}(ff^*)$ controls it uniformly. Take
-$nto∞$ then $lto∞$. The same argument with inputs reversed
+These approximants converge in the link form against a fixed Hilbert b:
+first replace b by bounded b_l; then (eq:links-relative-form)
+bounds the error by d|b_l|_∞^2|f_n-f|_2^2 on the tail while
+E_{N_0}((f_n-f)(f_n-f)^*)le4E_{N_0}(ff^*) controls it uniformly. Take
+nto∞ then lto∞. The same argument with inputs reversed
extends all bounded Gram identities to the stated domains.
Finally, if a gap event annihilates every one-column input, first
@@ -2108,12 +2107,12 @@ A fixed Fourier shift gives precisely the limiting intertwining identity
used in Lemma~lem:links-gram-transport; the finite-Fourier proof is
unchanged. Base decomposability and commutation with companions follow
before limits. An invariant one-column test gives an element of
-$mathcal N$;calN$; face-parabolic invariance, followed by the active-factor
+mathcal N;calN; face-parabolic invariance, followed by the active-factor
radical argument and the inactive-factor centralizer argument of
-Proposition~prop:links-link, puts it in $mathcal N^G=calN^G=mathbb C1$.bbC1$.
+Proposition~prop:links-link, puts it in mathcal N^G=calN^G=mathbb C1.bbC1.
For mixed links, independent target invariance puts the respective slices
-in $mathcal Nbar$calNbar$ and $mathcal N$;calN$; independent
+in mathcal NbarcalNbar and mathcal N;calN; independent
contracted-radical invariance makes them scalar, and polarization and
normal slice separation give the scalar joint effect.
@@ -2165,9 +2164,9 @@ imposes no new mixing or scalar-commutant assertion.
Proof.
-Write the source multiplier as $eta$. The regular representation with
-multiplier $overlineeta$, induced with the same section, has the conjugate
-of the scalar defect of $X_a(y)=pi_0(c(a,y))$, so the product of the two
+Write the source multiplier as eta. The regular representation with
+multiplier overlineeta, induced with the same section, has the conjugate
+of the scalar defect of X_a(y)=pi_0(c(a,y)), so the product of the two
fibre transporters satisfies the exact group composition law, and the
section continuity-in-measure argument gives strong continuity of the
resulting induced representation.
@@ -2180,7 +2179,7 @@ the two inputs. The generator identification of minimal spectral dilations
gives the enlarged dilation. A face spectral measure invariant under the
projective transport now commutes with a genuine source representation and
can be disintegrated with it. All mixing and scalarity statements continue
-to refer to $mathcal N$,calN$, not the enlarged commutant.
+to refer to mathcal N,calN, not the enlarged commutant.
## Escape from bounded target coordinates
@@ -2194,41 +2193,41 @@ measurement and for bounded pairs in the link.
Proof.
We record the finite-module Bessel estimate used here. For a bounded column
-$bin eL^2(B)^k$ and a Hilbert column $xi$, Parseval in regular coordinates
+bin eL^2(B)^k and a Hilbert column xi, Parseval in regular coordinates
gives
-$sum_{hin R}|< bw_h^*,xi>|^2=|b^*xi|_2^2le|b|_∞^2|xi|_2^2$,
+sum_{hin R}|< bw_h^*,xi>|^2=|b^*xi|_2^2le|b|_∞^2|xi|_2^2,
projective phases being harmless. The same holds for subsets, in particular
-$R'_j$-translates, and in finite matrix versions summing over fixed row and
+R'_j-translates, and in finite matrix versions summing over fixed row and
column indices.
For bounded end and start columns, the squared norm of the coefficient at a
-fixed measured index $ginGamma_j$ in $b(ay)^*X_a(y)f(y)$ is the sum over
-$R'_j$ of squared frame coefficients of the pair; equivalently these are
-inner products of $X_a(y)f(y)$ with partial translates of $b(ay)$, with $g$
+fixed measured index ginGamma_j in b(ay)^*X_a(y)f(y) is the sum over
+R'_j of squared frame coefficients of the pair; equivalently these are
+inner products of X_a(y)f(y) with partial translates of b(ay), with g
absorbed into one column. Use now the complementary regular realization for
-$Gamma_0times R'_j$. Project each column to a finite $Gamma_0$-coordinate
-window; these projections commute with partial $R'_j$-translates, preserving
+Gamma_0times R'_j. Project each column to a finite Gamma_0-coordinate
+window; these projections commute with partial R'_j-translates, preserving
Bessel bounds. Source transport is unitary and commutes with partial
translates, and a fixed measured shift preserves their bounds. Both
projected columns converge in integrated Hilbert norm as windows increase,
and the Bessel estimate on the other column bounds the sum of squared
-coefficient errors by that Hilbert error. Reindexing the end column at $ay$
+coefficient errors by that Hilbert error. Reindexing the end column at ay
does not change its averaged Hilbert error.
For fixed finite windows the translated supports are disjoint unless the
-source cocycle $c(a,y)$ belongs to a fixed finite subset of $Gamma_0$. For
-any $delta>0$, choose a compact set containing $d_0(y)$ outside a set of
-measure $delta$; by invariance, the same holds for $d_0(ay)$. On the
-intersection of the two good sets, $c(a,y)$ in a fixed finite set forces
-$a=d_0(ay)c(a,y)d_0(y)^{-1}$ to lie in a fixed compact subset of $G$,
+source cocycle c(a,y) belongs to a fixed finite subset of Gamma_0. For
+any delta>0, choose a compact set containing d_0(y) outside a set of
+measure delta; by invariance, the same holds for d_0(ay). On the
+intersection of the two good sets, c(a,y) in a fixed finite set forces
+a=d_0(ay)c(a,y)d_0(y)^{-1} to lie in a fixed compact subset of G,
impossible eventually on an escaping path. Exceptional measure is at most
-$2delta$, arbitrary.
+2delta, arbitrary.
Off that set, every partial-translate coefficient of the finite-window pair
vanishes; on it, their summed squares are uniformly bounded by Bessel and
column bounds. Integration gives convergence to zero, and the finite-window
approximation is removed afterward. We have proved vanishing of the norm at
-every fixed measured $Gamma_j$-index.
+every fixed measured Gamma_j-index.
A bounded Cartan set is compact in the whole ambient product, hence contains
only finitely many lattice elements. Summing gives zero link mass on each
@@ -2242,52 +2241,52 @@ one-column assertion.
[Full-flag obstruction]
-Let $S$ be the root-generated group of a noncompact isotropic almost simple
+Let S be the root-generated group of a noncompact isotropic almost simple
component of a semisimple source Levi subgroup, using its root-generated
lift from Convention~conv:setup-panels when the ambient action requires
-it. There is no nonzero positive map from $C(Z)$ for the full target flag
-space $Z$ to $mathcal A_j$calA_j$ which is target-equivariant and fixed by the
-source action $sigma|_S$.
+it. There is no nonzero positive map from C(Z) for the full target flag
+space Z to mathcal A_jcalA_j which is target-equivariant and fixed by the
+source action sigma|_S.
Proof.
-Suppose $Phi:C(Z)tomathcal A_j$calA_j$ is such a map. Its mass $a=Phi(1)$
-belongs to $mathcal N^S$:calN^S$: target equivariance makes it commute with the
-measured target action, and membership in $mathcal A_j$calA_j$ the rest. Choose
-$ε>0$ with $p=1_{[ε,∞)}(a)ne0$. On $pmathcal H$,calH$,
-$Psi(u)=pa^{-1/2}Phi(u)a^{-1/2}p$ is unital and completely positive (a
-positive map from $C(Z)$ is completely positive) with the same covariances.
-The normalization does not require $a$ to commute with $Phi(u)$.
-
-Choose a discrete free $Fcongmathbb F_2bbF_2subset S$, obtained from two
+Suppose Phi:C(Z)tomathcal A_jcalA_j is such a map. Its mass a=Phi(1)
+belongs to mathcal N^S:calN^S: target equivariance makes it commute with the
+measured target action, and membership in mathcal A_jcalA_j the rest. Choose
+ε>0 with p=1_{[ε,∞)}(a)ne0. On pmathcal H,calH,
+Psi(u)=pa^{-1/2}Phi(u)a^{-1/2}p is unital and completely positive (a
+positive map from C(Z) is completely positive) with the same covariances.
+The normalization does not require a to commute with Phi(u).
+
+Choose a discrete free Fcongmathbb F_2bbF_2subset S, obtained from two
sufficiently strong rank-one contractions with disjoint attracting and
-repelling neighborhoods. Let $mathcal D$calD$ be the concrete $C^*$-algebra on
-$pmathcal H$calH$ generated by cut base multipliers, the companion actions
-$W|_{R'_j}$, and $U_f$, $fin F$. These commute with the measured target
-action and every $Psi$ value.
+repelling neighborhoods. Let mathcal DcalD be the concrete C^*-algebra on
+pmathcal HcalH generated by cut base multipliers, the companion actions
+W|_{R'_j}, and U_f, fin F. These commute with the measured target
+action and every Psi value.
Use the minimal Stinespring construction with commutant lifting
[cite]. The minimal dilation has algebraic
-vectors $u ⊗ xi$ with form
-$<sum_iu_i ⊗ xi_i,sum_lv_l ⊗ eta_l>
-=sum_{i,l}<xi_i,Psi(u_ibarv_l)eta_l>$, and
-$widehatwidehat D(uD(u ⊗ xi)=u ⊗ Dxi$ satisfies
-$|widehatwidehat DD|le|D|$ in the concrete input norm. The covariant lift
-$T_g(u ⊗ xi)=(gcdot u) ⊗ W_gxi$ has the same multiplier
-$zeta$ as $W$.
+vectors u ⊗ xi with form
+<sum_iu_i ⊗ xi_i,sum_lv_l ⊗ eta_l>
+=sum_{i,l}<xi_i,Psi(u_ibarv_l)eta_l>, and
+widehatwidehat D(uD(u ⊗ xi)=u ⊗ Dxi satisfies
+|widehatwidehat DD|le|D| in the concrete input norm. The covariant lift
+T_g(u ⊗ xi)=(gcdot u) ⊗ W_gxi has the same multiplier
+zeta as W.
By cite[Proposition~2.5]{AnantharamanDelaroche2002}, the action of
-$Gamma_j$ on $Z$ is topologically amenable: choose continuous
-$eta_i:Ztoell^2(Gamma_j)$ with finite coordinate support, norms tending
-to $1$, and uniform asymptotic equivariance. Compactness of $Z$ makes their
+Gamma_j on Z is topologically amenable: choose continuous
+eta_i:Ztoell^2(Gamma_j) with finite coordinate support, norms tending
+to 1, and uniform asymptotic equivariance. Compactness of Z makes their
norms uniformly positive eventually, and normalizing pointwise gives
continuous unit fields with the same uniformity. In the dilation with
-spectral representation $pi_Z$ of $C(Z)$, put
-$Q_ieta=sum_{tinGamma_j}delta_t ⊗ pi_Z(xi_i(cdot)_t)eta$;
+spectral representation pi_Z of C(Z), put
+Q_ieta=sum_{tinGamma_j}delta_t ⊗ pi_Z(xi_i(cdot)_t)eta;
these are isometries intertwining the companion algebra exactly, with
-$|Q_iT_g-(lambda_g ⊗ T_g)Q_i|to0$ for each fixed $g$. Projective
-absorption $A(delta_t ⊗ eta)=delta_t ⊗ T_t^*eta$ conjugates
-$lambda_g ⊗ T_g$ to $lambda_zeta(g) ⊗ 1$, leaving the companion
+|Q_iT_g-(lambda_g ⊗ T_g)Q_i|to0 for each fixed g. Projective
+absorption A(delta_t ⊗ eta)=delta_t ⊗ T_t^*eta conjugates
+lambda_g ⊗ T_g to lambda_zeta(g) ⊗ 1, leaving the companion
unchanged. Taking limits on the original input gives, at every finite matrix
level,
@@ -2296,39 +2295,39 @@ level,
C^*_{r,zeta}(Gamma_j) ⊗ _{min}mathcal D}calD.
-The multiplier is that of $W$ and is conjugated in the right-action
+The multiplier is that of W and is conjugated in the right-action
convention.
-Complementary regularity makes $H$ a subrepresentation of a finite regular
-multiple for $Gamma_0times R'_j$ with split multiplier. Inducing the
-$Gamma_0$-coordinate and using $(y,h)mapsto d_0(y)h$ turns it into the
-$G$-coordinate, where $U_f$ acts as left translation times a scalar of
+Complementary regularity makes H a subrepresentation of a finite regular
+multiple for Gamma_0times R'_j with split multiplier. Inducing the
+Gamma_0-coordinate and using (y,h)mapsto d_0(y)h turns it into the
+G-coordinate, where U_f acts as left translation times a scalar of
modulus one. The companions use a separate twisted regular coordinate.
Tensoring with the measured regular representation in (eq:links-min-bound),
a single-base finite Fourier matrix field from the full split group algebra
-of $R$ acts fibrewise with its original matrix norm. Consequently, for
-contractions $C_f$, scalar phases $theta_f$, and a finitely supported
-probability $nu$ on $F$, the operator in this model has norm at most
-$|lambda_F(nu)|$: pointwise Hilbert norms are bounded by
-$sum_fnu(f)|eta(f^{-1}g)|$, and disintegrating Haar measure along left
-$F$-orbits gives the ordinary positive regular convolution bound on each
+of R acts fibrewise with its original matrix norm. Consequently, for
+contractions C_f, scalar phases theta_f, and a finitely supported
+probability nu on F, the operator in this model has norm at most
+|lambda_F(nu)|: pointwise Hilbert norms are bounded by
+sum_fnu(f)|eta(f^{-1}g)|, and disintegrating Haar measure along left
+F-orbits gives the ordinary positive regular convolution bound on each
orbit.
-We contradict this on one matrix vector. Regard $p$ as $k$ columns and put
-$T_f(y)=p(x)X_f(x)^*p(y)$ with $x=f^{-1}y$; this is a contraction. Since
-$p$ is $sigma_f$-invariant, $X_f(x)p(x)X_f(x)^*=p(y)$, and right
+We contradict this on one matrix vector. Regard p as k columns and put
+T_f(y)=p(x)X_f(x)^*p(y) with x=f^{-1}y; this is a contraction. Since
+p is sigma_f-invariant, X_f(x)p(x)X_f(x)^*=p(y), and right
multiplication after the shift gives
-$(R_{T_f}U_f^{(k)}p)(y)=X_f(x)p(x)cdot p(x)X_f(x)^*p(y)=p(y)$. For the
-finitely many $f$ in the support of $nu$, choose contraction fields
-$T_{f,n}$ simple in the base and finite in Fourier support with
-$T_{f,n}to T_f$ in integrated tracial $L^2$ (approximate in the unit ball
+(R_{T_f}U_f^{(k)}p)(y)=X_f(x)p(x)cdot p(x)X_f(x)^*p(y)=p(y). For the
+finitely many f in the support of nu, choose contraction fields
+T_{f,n} simple in the base and finite in Fourier support with
+T_{f,n}to T_f in integrated tracial L^2 (approximate in the unit ball
by simple-base fields, then use Kaplansky density and Fourier polynomials).
-Then $S_n=sum_fnu(f)R_{T_{f,n}}U_f^{(k)}$ has $S_nEto Ene0$. By
+Then S_n=sum_fnu(f)R_{T_{f,n}}U_f^{(k)} has S_nEto Ene0. By
(eq:links-min-bound) and the spatial estimate,
-$|S_n|le|lambda_F(nu)|$; for the uniform walk on two free generators
-and their inverses this is at most $sqrt3/2<1$ by Kesten's bound
+|S_n|le|lambda_F(nu)|; for the uniform walk on two free generators
+and their inverses this is at most sqrt3/2<1 by Kesten's bound
cite[Theorem~3]{Kesten1959}. The conclusions are incompatible since
-$0<|E|_2^2=mathsf T(p)<∞$.sfT(p)<∞$.
+0<|E|_2^2=mathsf T(p)<∞.sfT(p)<∞.
## Strict growth under a slower Levi translation
@@ -2336,145 +2335,145 @@ $0<|E|_2^2=mathsf T(p)<∞$.sfT(p)<∞$.
[A slower Levi translation removes every compact window]
Let an inner source face path have a positive target gap stratum
-$Ω_D$. After the inner limit, append a slower commuting split
-translation whose projection escapes in a noncompact simple component $S$ of
-the remaining source Levi; additional split parts commuting with $S$ are
-allowed. Every fixed compact window in $Ω_D$ has mass tending to zero
-in the outer limit. The entire positive contribution of $Ω_D$ then
-passes to strata $Ω_{D'}$ with $D'⊋ D$.
+Ω_D. After the inner limit, append a slower commuting split
+translation whose projection escapes in a noncompact simple component S of
+the remaining source Levi; additional split parts commuting with S are
+allowed. Every fixed compact window in Ω_D has mass tending to zero
+in the outer limit. The entire positive contribution of Ω_D then
+passes to strata Ω_{D'} with D'⊋ D.
Proof.
-The scalar mass $m_D>0$ is given by Proposition~prop:links-link;
+The scalar mass m_D>0 is given by Proposition~prop:links-link;
divide the stratum measurement by it, giving a unital full-data
-measurement $Psi$. If $D=Pi$, its image measurement is a nonzero
-full-flag map fixed by the remaining source component $S$, contradicting
-Lemma~lem:links-obstruction. Thus $DnePi$.
-
-The marginal inclusion. Let $Psi(F)=V_Ω^*M_FV_Ω$ be a
-minimal full-data dilation with image-flag map $x:Ω_Dto X_D$ and
-minimal image dilation $V:mathcal HcalHtomathcal K_X$.calK_X$. The generator map
-$M_uVfmapsto M_{ucirc x}V_Ω f$ preserves inner products and extends
-to an isometric inclusion $i:mathcal K_XcalK_Xtomathcal K_Ω$calK_Ω$ with
-$iV=V_Ω$, intertwining the image spectral measure, base multiplication,
+measurement Psi. If D=Pi, its image measurement is a nonzero
+full-flag map fixed by the remaining source component S, contradicting
+Lemma~lem:links-obstruction. Thus DnePi.
+
+The marginal inclusion. Let Psi(F)=V_Ω^*M_FV_Ω be a
+minimal full-data dilation with image-flag map x:Ω_Dto X_D and
+minimal image dilation V:mathcal HcalHtomathcal K_X.calK_X. The generator map
+M_uVfmapsto M_{ucirc x}V_Ω f preserves inner products and extends
+to an isometric inclusion i:mathcal K_XcalK_Xtomathcal K_ΩcalK_Ω with
+iV=V_Ω, intertwining the image spectral measure, base multiplication,
companions, and target lifts. Disintegrate
-$mathcal K_X=calK_X=int^ ⊕ K_x,dθ(x)$ with isometric fibre
-inclusions $i_x$.
-
-Positive density forms. Let $mathcal F_x$calF_x$ be the compact
-full-refinement space of the image face $x$, with compact-rotation
-probability $eta_x$. The residual projectivity of a full datum
-$omegainΩ_D$ transports that probability to a probability
-$k_omega(l),deta_x(l)$ on $mathcal F_x$,calF_x$, positive with total mass one.
-On a compact data window $CsubsetΩ_D$, this density has bounds
-$0<c_Cle k_omega(l)le C_C<∞$ uniformly in $omegain C$ and
-$linmathcal F_x$,calF_x$, because on such a window the residual transformations
+mathcal K_X=calK_X=int^ ⊕ K_x,dθ(x) with isometric fibre
+inclusions i_x.
+
+Positive density forms. Let mathcal F_xcalF_x be the compact
+full-refinement space of the image face x, with compact-rotation
+probability eta_x. The residual projectivity of a full datum
+omegainΩ_D transports that probability to a probability
+k_omega(l),deta_x(l) on mathcal F_x,calF_x, positive with total mass one.
+On a compact data window CsubsetΩ_D, this density has bounds
+0<c_Cle k_omega(l)le C_C<∞ uniformly in omegain C and
+linmathcal F_x,calF_x, because on such a window the residual transformations
and inverses range in a compact set, and their Jacobians against
-$eta_x$ are positive and bounded; at a finite place use analytic Haar
+eta_x are positive and bounded; at a finite place use analytic Haar
densities in finitely many compact-open charts.
-For fixed $(x,l)$, define bounded increasing
-$D_{n,x}(l)=i_x^*M_{min(k_omega(l),n)}i_x$. The limiting quadratic form
-$q_{x,l}(v)=|M_{√(k_omega(l))}i_xv|^2$ is closed. Its domain is dense
-for almost every $(x,l)$: choose $f_r$ with $Vf_r(x)$ total in $K_x$
+For fixed (x,l), define bounded increasing
+D_{n,x}(l)=i_x^*M_{min(k_omega(l),n)}i_x. The limiting quadratic form
+q_{x,l}(v)=|M_{√(k_omega(l))}i_xv|^2 is closed. Its domain is dense
+for almost every (x,l): choose f_r with Vf_r(x) total in K_x
almost everywhere (the measurable projection onto the fibrewise orthogonal
-complement kills angular multiples of $Vmathcal H$,calH$, so vanishes by
-minimality), then Tonelli and $int k,deta=1$ give
+complement kills angular multiples of Vmathcal H,calH, so vanishes by
+minimality), then Tonelli and int k,deta=1 give
[
int_{X_D}int_{mathcal F_x}calF_xq_{x,l}(Vf_r(x)),deta_x(l)dθ(x)
=|f_r|^2.
]
-For the associated positive $D_x(l)$, measurability follows from the
-bounded approximants, and $int_{mathcal F_x}calF_xq_{x,l}(v),deta_x(l)=|v|^2$
-for $vin K_x$. Consequently
+For the associated positive D_x(l), measurability follows from the
+bounded approximants, and int_{mathcal F_x}calF_xq_{x,l}(v),deta_x(l)=|v|^2
+for vin K_x. Consequently
(Tf)(x,l)=D_x(l)^{1/2}(Vf)(x)
-is a bounded isometry from $mathcal H$calH$ to
-$mathcal R=calR=int^ ⊕ _{X_D}L^2(mathcal F_x,calF_x,eta_x;K_x)dθ(x)$.
-
-Equivariance. For a target $gamma$, set
-$r_{gamma,x}(l')=d(gamma_*eta_x)/deta_{gamma x}(l')$. Then
-$k_{gammaomega}(gamma l)=r_{gamma,x}(gamma l)k_omega(l)$, and since
-$i$ intertwines target lifts, the compressed form is
-$D_{gamma x}(gamma l)=r_{gamma,x}(gamma l)u_{gamma,x}D_x(l)u_{gamma,x}^*$
-for the fibre-unitary $u_{gamma,x}$; the target lift on $mathcal R$calR$
-includes $r_{gamma,x}^{1/2}$ in addition to the image direct integral, so
-$T$ intertwines the target action. Base and companion operators commute
-with full-data multiplication and intertwine $i$, so $T$ intertwines them
-and all of $W_R$.
-
-For a source centralizer $s$, the invariant image marginal gives a
+is a bounded isometry from mathcal HcalH to
+mathcal R=calR=int^ ⊕ _{X_D}L^2(mathcal F_x,calF_x,eta_x;K_x)dθ(x).
+
+Equivariance. For a target gamma, set
+r_{gamma,x}(l')=d(gamma_*eta_x)/deta_{gamma x}(l'). Then
+k_{gammaomega}(gamma l)=r_{gamma,x}(gamma l)k_omega(l), and since
+i intertwines target lifts, the compressed form is
+D_{gamma x}(gamma l)=r_{gamma,x}(gamma l)u_{gamma,x}D_x(l)u_{gamma,x}^*
+for the fibre-unitary u_{gamma,x}; the target lift on mathcal RcalR
+includes r_{gamma,x}^{1/2} in addition to the image direct integral, so
+T intertwines the target action. Base and companion operators commute
+with full-data multiplication and intertwine i, so T intertwines them
+and all of W_R.
+
+For a source centralizer s, the invariant image marginal gives a
canonical lift on its minimal dilation defined by
-$U_s^X(M_uVf)=M_uVU_sf$; extend it to $widetilde U_s$ on $mathcal R$calR$ by
-its action on $K_x$ with induced base reindexing and identity on $l$. These
+U_s^X(M_uVf)=M_uVU_sf; extend it to widetilde U_s on mathcal RcalR by
+its action on K_x with induced base reindexing and identity on l. These
lifts use only the image marginal.
-The finite-trace intertwiner space. Let $mathscr E_0$scrE_0$ be the bounded
-intertwiners $mathcal HcalHtomathcal R$calR$ for the base and target actions;
-$B^*Cinmathcal N$calN$ for $B,Cinmathscr E_0$,scrE_0$, and
-$< B,C>_{mathscr E}scrE=tau_{mathcal N}calN(B^*C)$ completes to
-$mathscr E$scrE$ with
+The finite-trace intertwiner space. Let mathscr E_0scrE_0 be the bounded
+intertwiners mathcal HcalHtomathcal RcalR for the base and target actions;
+B^*Cinmathcal NcalN for B,Cinmathscr E_0,scrE_0, and
+< B,C>_{mathscr E}scrE=tau_{mathcal N}calN(B^*C) completes to
+mathscr EscrE with
|B|_{mathscr E}scrE^2=frac1dsum_{r=1}^k|Bb_r|^2,
- qquad
+
|Bf|^2le d|f|_∞^2|B|_{mathscr E}scrE^2,
-the second by testing $B^*B$ against the positive matrix $ff^*$. Source
-conjugation $mathscr U_sB=scrU_sB=widetilde U_sBU_s^*$ is isometric because
-$sigma_s$ preserves $tau_{mathcal N}calN$, and base defects cancel since $B$
+the second by testing B^*B against the positive matrix ff^*. Source
+conjugation mathscr U_sB=scrU_sB=widetilde U_sBU_s^* is isometric because
+sigma_s preserves tau_{mathcal N}calN, and base defects cancel since B
intertwines the base algebra; strong continuity follows from the frame
-equality and the two source lifts. The representation has no $S$-invariant
+equality and the two source lifts. The representation has no S-invariant
vector: if it had, a bounded intertwiner whose projection onto the
invariant subspace is nonzero has orbit convex hull containing that
-projection, and finite convex combinations $B_n$ with
-$|B_n|_{op}le|B|_{op}$ converge in $mathscr E$;scrE$; the
+projection, and finite convex combinations B_n with
+|B_n|_{op}le|B|_{op} converge in mathscr E;scrE; the
bounded-column estimate gives strong convergence on bounded columns, and
-$B_nto B_∞$ with $B_∞$ intertwining base and target and $S$-fixed.
-Then $umapsto B_∞^*M_{u(l)}B_∞$ is a nonzero positive full-flag
-map into $mathcal A_j$,calA_j$, target-equivariant and $S$-invariant, contradicting
+B_nto B_∞ with B_∞ intertwining base and target and S-fixed.
+Then umapsto B_∞^*M_{u(l)}B_∞ is a nonzero positive full-flag
+map into mathcal A_j,calA_j, target-equivariant and S-invariant, contradicting
Lemma~lem:links-obstruction.
-Howe--Moore now gives weak decay of translates of $T$ along escape in $S$,
+Howe--Moore now gives weak decay of translates of T along escape in S,
also when multiplied by commuting translations. Disintegrating into
-type-I irreducibles of $S$ (none trivial), commuting unitaries act on
+type-I irreducibles of S (none trivial), commuting unitaries act on
multiplicity spaces, and approximation by finite tensor sums reduces to
-matrix coefficients of the irreducible $S$-representations, tending to
+matrix coefficients of the irreducible S-representations, tending to
zero uniformly over the multiplicity unitaries; dominated convergence
finishes.
The positive compact-window comparison. For a compact window
-$CsubsetΩ_D$, let $L_C=i^*M_{1_C}i$; it is decomposable and positive,
-$V^*L_CV=Psi(C)$. The density lower bound gives $D_x(l)ge c_CL_C(x)$ in
+CsubsetΩ_D, let L_C=i^*M_{1_C}i; it is decomposable and positive,
+V^*L_CV=Psi(C). The density lower bound gives D_x(l)ge c_CL_C(x) in
form order, and operator monotonicity of square roots yields
D_x(l)^{1/2}ge√(c_C),L_C(x).
-Write the extended path $a_tb_s$, inner $t$-limit first. The exact cocycle
-formula and change of base $z=b_sy$ show the inner full-data law is
-precomposed by $U_{b_s}$: $Psi_s(F)=U_{b_s}^*Psi(F)U_{b_s}$, with
-unchanged image marginal. Set $T^{(s)}=widetilde U_{b_s}^*TU_{b_s}$. The
-marginal lift fixes $(x,l)$, so its fibre expression is
-$(U_{b_s}^X)^*D_x(l)^{1/2}U_{b_s}^X$ applied to $Vf$; inequality
-(eq:links-positive-window) persists. With $J_0f(x,l)=Vf(x)$,
-$Bmapsto< Bf,J_0f>$ is continuous on $mathscr E$scrE$ by
+Write the extended path a_tb_s, inner t-limit first. The exact cocycle
+formula and change of base z=b_sy show the inner full-data law is
+precomposed by U_{b_s}: Psi_s(F)=U_{b_s}^*Psi(F)U_{b_s}, with
+unchanged image marginal. Set T^{(s)}=widetilde U_{b_s}^*TU_{b_s}. The
+marginal lift fixes (x,l), so its fibre expression is
+(U_{b_s}^X)^*D_x(l)^{1/2}U_{b_s}^X applied to Vf; inequality
+(eq:links-positive-window) persists. With J_0f(x,l)=Vf(x),
+Bmapsto< Bf,J_0f> is continuous on mathscr EscrE by
(eq:links-intertwiner-frame). Weak decay and the positive inequality
give
-$0le√(c_C)< f,Psi_s(C)f>le< T^{(s)}f,J_0f>to0$;
-density of bounded columns and $0lePsi_s(C)le1$ extend the vanishing to
+0le√(c_C)< f,Psi_s(C)f>le< T^{(s)}f,J_0f>to0;
+density of bounded columns and 0lePsi_s(C)le1 extend the vanishing to
every input. This proves disappearance of the actual translated window
mass.
-The entire stratum contribution. Each $Psi_s$ has mass one on
-$Ω_D$. Take its outer limit on $Ω_Dbar$. Every continuous
-test compactly supported in $Ω_D$ has zero limiting effect by the
+The entire stratum contribution. Each Psi_s has mass one on
+Ω_D. Take its outer limit on Ω_Dbar. Every continuous
+test compactly supported in Ω_D has zero limiting effect by the
window estimate, so a countable exhaustion shows the outer law gives no
-mass to $Ω_D$. Lemma~lem:links-matrix-data places the remaining
-support in $Ω_Dbar∖Ω_D
-⊆∪_{D'⊋ D}Ω_{D'}$. Restoring $m_D$ proves the
+mass to Ω_D. Lemma~lem:links-matrix-data places the remaining
+support in Ω_Dbar∖Ω_D
+⊆∪_{D'⊋ D}Ω_{D'}. Restoring m_D proves the
assertion for the entire positive contribution; further subpieces are
dominated by the positive total measurements.
@@ -2484,9 +2483,9 @@ dominated by the positive total measurements.
[Gap count and vertex quotients]
In one measured copy, every positive stratum of a source path toward a face
-of type $I$ has at most $|I|$ infinite gaps. A vertex path has exactly one
+of type I has at most |I| infinite gaps. A vertex path has exactly one
infinite gap. In a successive single-gap lex chain starting at a vertex,
-the stage with $k$ source gaps has exactly $k$ target gaps, so each such
+the stage with k source gaps has exactly k target gaps, so each such
extension adds exactly one.
For a quotient of a faster and a slower vertex translation in a common
@@ -2496,29 +2495,29 @@ if they are not.
Proof.
-Put $r=|Pi|$ and start with a positive target stratum $D$ on a face of
-type $I$. Append the $r-|I|$ missing source coordinates at successively
+Put r=|Pi| and start with a positive target stratum D on a face of
+type I. Append the r-|I| missing source coordinates at successively
slower scales. At each stage the fundamental split vector for a missing
root has a nonzero projection to its remaining-Levi component, whose other
split parts commute; so Lemma~lem:links-strict-growth applies. If a
full target stratum were reached before the source face were complete, the
obstruction in that lemma would already contradict. Otherwise the tracked
contribution passes through a strictly increasing chain of length
-$r-|I|$; since $D$ has at most $r$ types, $|D|+r-|I|le r$, proving
-$|D|le|I|$. If $I=Pi$, this is the automatic ceiling and no
+r-|I|; since D has at most r types, |D|+r-|I|le r, proving
+|D|le|I|. If I=Pi, this is the automatic ceiling and no
remaining-Levi obstruction is invoked.
For a vertex, Lemma~lem:links-no-bounded-stratum excludes zero gaps
and the upper bound excludes two or more. The same strict-growth argument
-applied successively from the vertex gives at least $k$ gaps at stage $k$,
-while the upper bound gives at most $k$.
+applied successively from the vertex gives at least k gaps at stage k,
+while the upper bound gives at most k.
Finally choose the common apartment and let the faster vertex be a dominant
-fundamental ray $v$ of type $j$. If the slower vertex is coaxial, the
+fundamental ray v of type j. If the slower vertex is coaxial, the
quotient is again a vertex translation at the faster scale (possibly with
-opposed orientation), and the vertex conclusion applies. If its vector $w$
+opposed orientation), and the vertex conclusion applies. If its vector w
is not coaxial, the projection to a simple factor of the remaining Levi is
-nonzero, since the common kernel of $Pi∖{j}$ is $mathbb Rv$.bbRv$.
+nonzero, since the common kernel of Pi∖{j} is mathbb Rv.bbRv.
Choosing a dominant chamber inside the remaining Levi, the slower quotient
component escapes in a noncompact simple factor with other split parts
commuting; strict growth from the one-gap faster limit forces at least one
@@ -2530,25 +2529,25 @@ limits.
We continue with the induced finite-module realization of
Section~sec:links. All target indices here are indices of a measured
-copy of $Gamma$; the comparison group $Λ$ remains in the coefficient
-algebra, and no flag space carries an action of $Λ$. Our objective is
+copy of Gamma; the comparison group Λ remains in the coefficient
+algebra, and no flag space carries an action of Λ. Our objective is
a sharp flag measurement at each source flag, with an exact equation for
the residual projectivity on a missing panel.
## A transverse product estimate
-For a scalar function $f$ of the measured Fourier index, let $D_f$ denote
-diagonal multiplication of the Fourier array; it is an $L^2$ operator, and
+For a scalar function f of the measured Fourier index, let D_f denote
+diagonal multiplication of the Fourier array; it is an L^2 operator, and
we do not assume it preserves operator-bounded columns. We estimate
(D_g B_0)(D_f A_0),qquadwith index product qp,
-where $A_0,B_0$ are operator-bounded with Fourier expansions
-$A_0=sum_pa_pw_p$, $B_0=sum_qb_qw_q$, operator-valued coefficients in the
-unmeasured algebra, $p$ an index of $A_0$ and $q$ an index of $B_0$.
-Matrix sizes are fixed during every limit; a trace of mass $t$ rescales an
-operator-norm-only $L^2$ bound by $sqrt t$, but the homogeneous estimate
+where A_0,B_0 are operator-bounded with Fourier expansions
+A_0=sum_pa_pw_p, B_0=sum_qb_qw_q, operator-valued coefficients in the
+unmeasured algebra, p an index of A_0 and q an index of B_0.
+Matrix sizes are fixed during every limit; a trace of mass t rescales an
+operator-norm-only L^2 bound by sqrt t, but the homogeneous estimate
below uses the actual Hilbert norms. Rectangular matrices are included by
embedding in one fixed square matrix algebra and using the corresponding
corners.
@@ -2556,23 +2555,23 @@ corners.
[Transverse products]
Fix face types, a thickness bound, and sufficiently small compact
-transverse face patches at the basepoint $o$. Suppose the supports of $f$
-and $g$ lie within that thickness of the respective positive cones, with
+transverse face patches at the basepoint o. Suppose the supports of f
+and g lie within that thickness of the respective positive cones, with
all specified positive gaps sufficiently large, and that the ideal
-directions of $po$ and $q^{-1}o$ belong to the fixed transverse patches.
-Then $mu(qp)=mu(q)+mu(p)+O(1)$, uniformly on the supports. The product
-in (eq:input-2) has a uniform tracial $L^2$ bound in either of the
+directions of po and q^{-1}o belong to the fixed transverse patches.
+Then mu(qp)=mu(q)+mu(p)+O(1), uniformly on the supports. The product
+in (eq:input-2) has a uniform tracial L^2 bound in either of the
following situations:
[label=(roman*)]
-item For each coordinate contributed by both cones, one of its two
+ For each coordinate contributed by both cones, one of its two
supports is restricted to a moving interval of a fixed bounded width. The
separate cutoff factors may be arbitrary uniformly bounded functions.
-item One of the face types is a single vertex. The cutoff factors are
+ One of the face types is a single vertex. The cutoff factors are
smooth with uniformly bounded sufficiently high seminorms in fixed compact
normalized matrix charts; at finite places this means locally constant with
a uniformly bounded level.
-Write $f=h_Af_1$, $g=h_Bg_1$. In case (i), all four factors may be
+Write f=h_Af_1, g=h_Bg_1. In case (i), all four factors may be
arbitrary uniformly bounded scalar functions; in case (ii), each factor
has the smooth seminorm bounds or fixed clopen level specified. Then
@@ -2582,30 +2581,30 @@ has the smooth seminorm bounds or fixed clopen level specified. Then
The constant includes the bounds on the scalar factors. The same estimate
-holds for a joint symbol $sum_{q,p}h_B(q)h_A(p)m(q,p)b_qa_pw_qw_p$, where
-$w_qw_p$ is the actual twisted monomial product and $m$ is a smooth joint
+holds for a joint symbol sum_{q,p}h_B(q)h_A(p)m(q,p)b_qa_pw_qw_p, where
+w_qw_p is the actual twisted monomial product and m is a smooth joint
function (or finite clopen level at finite places) with uniformly bounded
derivatives. Consequently an additional smooth cutoff factor whose
one-column Hilbert norm tends to zero makes the product tend to zero,
provided the other bounds and the other cut Hilbert norm remain bounded;
-for a fixed finite trace of mass $t$,
-$|(D_gB_0)(D_fA_0)|_2le Csqrt t|A_0|_∞|B_0|_∞$.
+for a fixed finite trace of mass t,
+|(D_gB_0)(D_fA_0)|_2le Csqrt t|A_0|_∞|B_0|_∞.
Proof.
Common apartments and addition. On compact transverse patches, the
two ideal faces admit common apartment or flat sections with origins in a
-bounded set. After erasing the bounded thickness, $po=z+A+O(1)$ and
-$q^{-1}o=z-B+O(1)$ with $A,B$ positive and $z$ bounded, so the vector
-distance is $A+B+O(1)$; the vector-distance inequality under bounded
+bounded set. After erasing the bounded thickness, po=z+A+O(1) and
+q^{-1}o=z-B+O(1) with A,B positive and z bounded, so the vector
+distance is A+B+O(1); the vector-distance inequality under bounded
endpoint perturbations gives the Cartan addition. This uses the whole
ambient product and no discreteness of a proper projection.
We need Lipschitz stability of interpolation: from a fixed endpoint,
-multiply the simple coordinates $L_i$ of a dominant vector by $t_iin[0,1]$.
+multiply the simple coordinates L_i of a dominant vector by t_iin[0,1].
In an archimedean polar chart, the angular differential in a positive-root
-direction is multiplied by $sinh(alpha(tcdot L))/sinh(alpha(L))le1$
-and the radial differential by the bounded linear map $Lmapsto tcdot L$;
+direction is multiplied by sinh(alpha(tcdot L))/sinh(alpha(L))le1
+and the radial differential by the bounded linear map Lmapsto tcdot L;
a wall stabilizer fixes the interpolated vector since zero simple
coordinates stay zero. Approximation of paths by regular paths gives
continuity across walls; reversing endpoints uses opposition and the
@@ -2613,10 +2612,10 @@ complementary fractions. For finite-place buildings, the same statement is
Lemma~lem:setup-building-interpolation. The product of real and
finite-place bounds covers the whole ambient space.
-The common shift. Suppose $qp=q'p'$ and $k=p(p')^{-1}$. With
-primed coordinates $z'+A',z'-B'$, Cartan addition gives $A+B=A'+B'+O(1)$.
-Interpolate from the negative endpoint with $t_i=B_i'/(A_i'+B_i')$ (any
-choice in $[0,1]$ if the denominator vanishes); these return $z'$ in the
+The common shift. Suppose qp=q'p' and k=p(p')^{-1}. With
+primed coordinates z'+A',z'-B', Cartan addition gives A+B=A'+B'+O(1).
+Interpolate from the negative endpoint with t_i=B_i'/(A_i'+B_i') (any
+choice in [0,1] if the denominator vanishes); these return z' in the
primed model, and Lipschitz equivariance gives
ko=z+B'-B+O(1).
@@ -2625,35 +2624,35 @@ primed model, and Lipschitz equivariance gives
The fractions are bounded and held fixed while applying the estimate.
In case (i), every overlapping coordinate has bounded variation on at
-least one side; the relation between $A+B$ and $A'+B'$ bounds the
-corresponding coordinate of $B'-B$, and outside the overlap one side is
-bounded by thickness. Thus $k$ lies in a fixed compact subset of the whole
-product $G$, whose intersection with $Gamma$ is finite, uniformly in the
+least one side; the relation between A+B and A'+B' bounds the
+corresponding coordinate of B'-B, and outside the overlap one side is
+bounded by thickness. Thus k lies in a fixed compact subset of the whole
+product G, whose intersection with Gamma is finite, uniformly in the
moving-width positions.
In case (ii), an unbounded difference can occur only in the single overlap
-coordinate $i$; otherwise we are in the bounded-shift case. Put
-$delta=B_i'-B_i$. Then $A_i-A_i'=delta+O(1)$ and
-$p=kp'$, $q'=qk$, $(q')^{-1}=k^{-1}q^{-1}$. Applying
+coordinate i; otherwise we are in the bounded-shift case. Put
+delta=B_i'-B_i. Then A_i-A_i'=delta+O(1) and
+p=kp', q'=qk, (q')^{-1}=k^{-1}q^{-1}. Applying
(eq:transfer-common-shift) to both orientations gives
-$ko=z+delta e_i+O(1)$ and $k^{-1}o=z'-delta e_i+O(1)$. With $U$ the
-patch of type-$i$ vertices incident to the positive endpoint patch and $V$
-the patch of type-$i^*$ vertices incident to the negative endpoint patch,
-and buffered larger patches $U^+,V^+$ with every pair opposite, for
-sufficiently large positive $delta$ the forward vertex of $k$ lies in
-$U^+$ and its reverse vertex in $V^+$, reversed for negative $delta$;
-the patches are disjoint in the visual boundary, so a fixed large $k$
-cannot have both orientations. Bounded $delta$ is in the bounded-shift
+ko=z+delta e_i+O(1) and k^{-1}o=z'-delta e_i+O(1). With U the
+patch of type-i vertices incident to the positive endpoint patch and V
+the patch of type-i^* vertices incident to the negative endpoint patch,
+and buffered larger patches U^+,V^+ with every pair opposite, for
+sufficiently large positive delta the forward vertex of k lies in
+U^+ and its reverse vertex in V^+, reversed for negative delta;
+the patches are disjoint in the visual boundary, so a fixed large k
+cannot have both orientations. Bounded delta is in the bounded-shift
case.
-In the positive orientation the shorter arguments are $p'$ and $q^{-1}$,
-with length-increasing substitutions $p'mapsto kp'$ and
-$q^{-1}mapsto k^{-1}q^{-1}$. The reverse vertex of the first multiplier is
-in $V^+$ (transverse to the entire buffered positive input patch); of the
-second, in $U^+$ (transverse to the negative input patch). For a fixed
-finite-dimensional representation, put $S(g)=R(g)/|R(g)|$. On either
-substitution $rmapsto ur$, transverse Cartan addition plus highest-weight
-comparison gives $|S(u)S(r)|ge c>0$ and $|S(r^{-1})S(u^{-1})|ge c>0$
+In the positive orientation the shorter arguments are p' and q^{-1},
+with length-increasing substitutions p'mapsto kp' and
+q^{-1}mapsto k^{-1}q^{-1}. The reverse vertex of the first multiplier is
+in V^+ (transverse to the entire buffered positive input patch); of the
+second, in U^+ (transverse to the negative input patch). For a fixed
+finite-dimensional representation, put S(g)=R(g)/|R(g)|. On either
+substitution rmapsto ur, transverse Cartan addition plus highest-weight
+comparison gives |S(u)S(r)|ge c>0 and |S(r^{-1})S(u^{-1})|ge c>0
uniformly; the inverse coordinates are separately normalized inverse
matrices, no inverse of a possibly singular limit. Compactness preserves
a common lower bound on fixed larger coordinate boxes, so normalized
@@ -2662,9 +2661,9 @@ finite place, maximal-coordinate ratio charts give uniformly nonzero
denominators.
Gram coefficients and twists. Expand the squared norm in
-(eq:input-2) and group pairs with $qp=q'p'$ by $k=p(p')^{-1}$. After
+(eq:input-2) and group pairs with qp=q'p' by k=p(p')^{-1}. After
trace cyclicity the summand is the pairing of two Gram coefficients, one
-made from $a_{p'}a_p^*$ and the other from $b_q^*b_{q'}$ (adjoints changed
+made from a_{p'}a_p^* and the other from b_q^*b_{q'} (adjoints changed
in the opposite orientation). The cocycle factors are exactly the factors
in these adjoint products; we do not replace the operator-valued
convolution by an absolute-value convolution. For bounded shifts, row and
@@ -2673,19 +2672,19 @@ norm by the appropriate uncut column norm squared; its trace norm is
bounded by the relevant cut Hilbert norm squared, and tracial
Cauchy--Schwarz gives (eq:transfer-homogeneous).
-For unbounded shifts, if $p=kp'$ is the length-increased side, retain
-$h_A(p')bar$ in $f(kp')f(p')bar=
-h_A(p')barf(kp')f_1(p')bar$ and expand the rest in the
-shorter argument $p'$. On the other side, retain $h_B(q)$ in
-$g(q)g(qk)bar=h_B(q)g_1(q)g(qk)bar$; its shorter geometric
-argument is $q^{-1}$. Conjugating Gram coefficients may conjugate these
+For unbounded shifts, if p=kp' is the length-increased side, retain
+h_A(p')bar in f(kp')f(p')bar=
+h_A(p')barf(kp')f_1(p')bar and expand the rest in the
+shorter argument p'. On the other side, retain h_B(q) in
+g(q)g(qk)bar=h_B(q)g_1(q)g(qk)bar; its shorter geometric
+argument is q^{-1}. Conjugating Gram coefficients may conjugate these
formulas, with no effect on the cut Hilbert norm; the opposite orientation
-interchanges primes, retaining $h_A(p)$ and $h_B(q')$. Extend with an
+interchanges primes, retaining h_A(p) and h_B(q'). Extend with an
initial-patch cutoff with slack. For the first Gram sequence, choose
conjugated basis functions so its full weight has the form
- w_k(r)=sum_nc_n(k)h(r)u_n(r)bar,qquad
- |u_n|_∞le1,qquad
+ w_k(r)=sum_nc_n(k)h(r)u_n(r)bar,
+ |u_n|_∞le1,
sum_nsup_k|c_n(k)|<∞.
@@ -2693,14 +2692,14 @@ The strong order of sum and supremum follows from Fourier expansion on a
slightly larger real box with uniform derivative bounds; at finite places
a finite clopen mask basis replaces it. Minkowski, Plancherel, and the
Hilbert ideal inequality give
-$|(sum_nc_n(k)[A_0(D_{hu_n}A_0)^*]_k)_k|_{ell^2(L^2)}
-le(sum_nsup_k|c_n(k)|)|A_0|_∞|D_hA_0|_2$.
+|(sum_nc_n(k)[A_0(D_{hu_n}A_0)^*]_k)_k|_{ell^2(L^2)}
+le(sum_nsup_k|c_n(k)|)|A_0|_∞|D_hA_0|_2.
The other side is symmetric. Cauchy--Schwarz in the shift variable then
proves (eq:transfer-homogeneous), and the small-factor assertion.
Smooth joint symbols follow from summable separated expansions retaining
the restrictions with slack; approximating bounded columns by uniformly
-bounded Fourier polynomials with unary $L^1$ convergence gives the asserted
-$L^2$ bound. No step requires operator-norm control of an angularly cut
+bounded Fourier polynomials with unary L^1 convergence gives the asserted
+L^2 bound. No step requires operator-norm control of an angularly cut
column.
@@ -2715,18 +2714,18 @@ exhausted only after the relevant inner limit.
## Fullness and the permutation of types
-For an archimedean target vertex module $V$, let $n(v)$ be a unit
-representative of its highest line; $n(v)n(v)^*$ is independent of the
+For an archimedean target vertex module V, let n(v) be a unit
+representative of its highest line; n(v)n(v)^* is independent of the
phase. We also allow the trivial one-dimensional module; at a finite target
place only this trivial-module version is used.
[Fullness of open patches]
-Let $Phi_z$ be a nonzero family of image measurements as in
+Let Phi_z be a nonzero family of image measurements as in
Proposition~prop:links-link, transported over its source label space
and normalized by its positive scalar stratum mass to be unital. For every
-nonempty open target patch $O$, the closed sum, over conull source labels
-$z$, of the ranges of
+nonempty open target patch O, the closed sum, over conull source labels
+z, of the ranges of
int_O n(v)n(v)^* ⊗ Phi_z(dv)
@@ -2737,34 +2736,34 @@ in the three-slot system.
Proof.
-Omit $O$ first. Source covariance makes the uncut range join a projection
-over $(mathcal A_1)^calA_1)^sigma$; target covariance transports its range by
+Omit O first. Source covariance makes the uncut range join a projection
+over (mathcal A_1)^calA_1)^sigma; target covariance transports its range by
finite-dimensional linear target matrices, unchanged by positive
-renormalization of $n(v)$. Inducing the target lattice action with its
+renormalization of n(v). Inducing the target lattice action with its
finite invariant trace gives an induced range field intertwining a
-trace-isometric ambient action with the inverse linear transport on $V$.
+trace-isometric ambient action with the inverse linear transport on V.
-Let a positive matrix $A$ have weight spaces with strictly increasing
-eigenvalues and let $P$ be a projection over a finite algebra. Filtering
-$ran P$ by its intersections with the weight filtration and taking the
+Let a positive matrix A have weight spaces with strictly increasing
+eigenvalues and let P be a projection over a finite algebra. Filtering
+ran P by its intersections with the weight filtration and taking the
successive leading projections onto the weight spaces gives, by polar
-decomposition, an orthogonal sum $Q$ with $tr(Q)=
-tr(P)$ and $ran Q$ strongly approximated by $A^nran P$.
+decomposition, an orthogonal sum Q with tr(Q)=
+tr(P) and ran Q strongly approximated by A^nran P.
Since the range projections have constant finite trace, they converge in
-trace $L^2$ to $Q$. Apply pointwise to the induced field; a convergent
+trace L^2 to Q. Apply pointwise to the induced field; a convergent
orbit of a trace-isometric action is fixed (the distance between
consecutive orbit points is constant and tends to zero). Invariance under
all positive split matrices forces the range to split off the whole matrix
factor; the remainder descends to the scalar joint fixed algebra. Thus
the uncut join is full.
-Restore $O$: the finite trace dimension of the patch join is constant
+Restore O: the finite trace dimension of the patch join is constant
along target lattice translates. For any finite scalar test measure on
-$Z$, choose a regular contracting direction with attractor in $O$ and
+Z, choose a regular contracting direction with attractor in O and
repeller transverse to almost every point of the measure; all-factor
mixing supplies lattice elements whose two compact frames approximate
the chosen frames along increasingly long translations, so suitable
-translates of $O$ exhaust the measure. Applied to a faithful
+translates of O exhaust the measure. Applied to a faithful
scalarization of a countable total family, the complements of the
translated patch joins have trace tending to zero, giving patch fullness.
Normal copies follow from normal ampliation in Proposition~prop:links-link.
@@ -2778,21 +2777,21 @@ strong continuity is used only for these constructed families.
[Type comparison]
-There is a unique permutation $p$ of the full relative type list $Pi$
-such that every candidate at source type $i$ takes values in target type
-$p(i)$; it preserves opposition, $p(i^*)=p(i)^*$. For noncoaxial source
+There is a unique permutation p of the full relative type list Pi
+such that every candidate at source type i takes values in target type
+p(i); it preserves opposition, p(i^*)=p(i)^*. For noncoaxial source
vertices of dual types, all ordered products of candidates vanish on
transverse target patches. The permutation preserves the collection of
isolated types.
Proof.
-Choose paths $t,a$ whose inverses tend to the two source vertices in a
-common apartment. Take successive limits, $t$ faster. Pairing the ordered
+Choose paths t,a whose inverses tend to the two source vertices in a
+common apartment. Take successive limits, t faster. Pairing the ordered
cut effects between bounded columns and using trace cyclicity gives a
-pairing of $(D_AX_tf)(D_BX_af')^*$ with the quotient-path matrix $X_tX_a^*$
-after base reindexing; the latter is the transporter of $ta^{-1}$ up to a
-common scalar, with index product $pq^{-1}$.
+pairing of (D_AX_tf)(D_BX_af')^* with the quotient-path matrix X_tX_a^*
+after base reindexing; the latter is the transporter of ta^{-1} up to a
+common scalar, with index product pq^{-1}.
If coaxial, the quotient path has exactly one target gap
(Proposition~prop:links-gap-count). Nondual transverse target types
@@ -2812,9 +2811,9 @@ Fullness gives nonzero ordered cut product for generic source pairs
(apply Lemma~lem:transfer-fullness successively). Generic dual
source vertices are opposite, hence coaxial; generic nondual source types
are noncoaxial. Thus candidates at dual source types have dual target
-types, and at nondual source types nondual target types. If $T_i$ is the
-possible target type set at source $i$, every $jin T_i$ and $kin
-T_{i^*}$ satisfy $k=j^*$. Since the $T_i$ are nonempty, each is a
+types, and at nondual source types nondual target types. If T_i is the
+possible target type set at source i, every jin T_i and kin
+T_{i^*} satisfy k=j^*. Since the T_i are nonempty, each is a
singleton, and the resulting map is injective and hence a permutation.
At higher-rank self-opposite source types, extend nonopposition by
@@ -2824,14 +2823,14 @@ and gallery connectivity would make the candidate family constant, giving
a source-invariant target-equivariant flag probability against contraction
(Lemma~lem:transfer-fullness). Thus self-opposite higher-rank types
do not map to isolated types; non-self-opposite ones do not either by
-opposition preservation. Since $p$ is a permutation, it permutes the
+opposition preservation. Since p is a permutation, it permutes the
isolated types.
## Equality at isolated types
Fix an isolated archimedean source type and its matched target boundary
-$Z$.
+Z.
[Disjoint angular forms]
@@ -2840,7 +2839,7 @@ different candidate families at the fixed isolated type, there is no
common nonzero target-equivariant positive map dominated by the two
candidate maps (complete-positive domination). After induction, there is
no common nonzero linear or conjugate-linear functional on
-$C^∞(Z)odotmathcal H$calH$ bounded by both angular form norms
+C^∞(Z)odotmathcal HcalH bounded by both angular form norms
b⟼(int_Z< b(v),Phi_x(dv)b(v)>)^{1/2}.
@@ -2851,40 +2850,40 @@ source covariance.
Proof.
Undo source induction: simultaneous covariance yields measurable unital
-CP maps $F_x:C(Z)to B(H)$ with
-$F_{gamma x}=pi_0(gamma)F_xpi_0(gamma)^*$, equivariant for $R$; the
+CP maps F_x:C(Z)to B(H) with
+F_{gamma x}=pi_0(gamma)F_xpi_0(gamma)^*, equivariant for R; the
assertion follows from the lattice cocycle formula or Haar
disintegration, scalar phases disappearing in conjugation. The source
lattice action on pairs of this boundary is ergodic (the open-pair
stabilizer contains a split subgroup escaping in the active factor and
all complementary factors, so all-factor mixing applies).
-Encode $F_x$ by its minimal covariant Stinespring dilation and the vector
-obtained from the embedded columns of $e$, giving a positive functional
-$phi_x$ on the separable matrix-amplified full twisted crossed product
-$mathfrak C$frakC$ with norm $d$. The group and matrix operators act on these
+Encode F_x by its minimal covariant Stinespring dilation and the vector
+obtained from the embedded columns of e, giving a positive functional
+phi_x on the separable matrix-amplified full twisted crossed product
+mathfrak CfrakC with norm d. The group and matrix operators act on these
columns by right multiplication; the regular-normal central summand of
the group algebra bidual supplies canonical right multipliers, so the
vector functional is supported on the corner of the corner, and the
-positive subfunctionals of $phi_x$ encode exactly the corresponding
+positive subfunctionals of phi_x encode exactly the corresponding
equivariant subordinate maps (covariant Arveson domination
cite[Theorem~1.4.2]{Arveson1969}).
-Fix the central support $z$ of one functional of the second family.
-$zmathfrak C^frakC^{**}$ has separable predual, and the components $phi_x^z$
+Fix the central support z of one functional of the second family.
+zmathfrak C^frakC^{**} has separable predual, and the components phi_x^z
form a measurable norm-separable family: test domination on a countable
norm-dense positive family, let
-$n(x)=sup{|theta|:0lethetalephi_x, theta is z-normal}$
+n(x)=sup{|theta|:0lethetalephi_x, theta is z-normal}
be upper semianalytic, and apply Jankov--von Neumann
-[cite] to obtain measurable dominated $theta_q(x)$ with
-norm $>q$ on ${n>q}$; rational $q_m(x)<n(x)$ with $n(x)-q_m(x)<1/m$
-gives $|phi_x^z-theta_{q_m(x)}(x)|<1/m$. No simultaneous measurable
+[cite] to obtain measurable dominated theta_q(x) with
+norm >q on {n>q}; rational q_m(x)<n(x) with n(x)-q_m(x)<1/m
+gives |phi_x^z-theta_{q_m(x)}(x)|<1/m. No simultaneous measurable
choice of second-family central supports is needed.
-The distance between $phi_x^z$ and $phi_y^z$ is invariant under the
+The distance between phi_x^z and phi_y^z is invariant under the
diagonal source action, hence essentially constant by double ergodicity.
A separable metric-valued measurable map cannot have strictly positive
-constant pair distance. So $phi_x^z$ is constant a.e.; if nonzero, the
+constant pair distance. So phi_x^z is constant a.e.; if nonzero, the
encoding gives a nonzero source-invariant, target-equivariant positive
map, whose invariant finite trace yields an invariant flag probability,
contradicting target contraction.
@@ -2896,7 +2895,7 @@ Fubini gives the first assertion for independent pairs.
Reinduce and consider (eq:input-3). A nonzero scalar functional
bounded by both gives the parallel sum of the two squared forms. Build it
as the quotient of the direct sum of the two completions by the span of
-pairs $(h,-h)$ from the common algebraic domain. This subspace is reducing
+pairs (h,-h) from the common algebraic domain. This subspace is reducing
for angular multiplication, base multipliers, and target covariances.
Angular multiplication descends; compression at constant input vectors
gives a nonzero decomposable equivariant CP map dominated by both
@@ -2908,8 +2907,8 @@ transports the generic conclusion over the single transverse orbit.
[A consequence for sharp measurements]
If the two measurements in Lemma~lem:transfer-disjointness are
-sharp, the ranges $P_x(U)P_y(V)mathcal H$calH$ for disjoint angular patches
-$U,V$ have dense sum. A vector orthogonal to all these ranges would,
+sharp, the ranges P_x(U)P_y(V)mathcal HcalH for disjoint angular patches
+U,V have dense sum. A vector orthogonal to all these ranges would,
by regular approximation, have the same spectral vector for the two
PVMs; pairing gives a common functional bounded by the two forms,
contrary to the lemma.
@@ -2922,32 +2921,32 @@ choices agree.
Proof.
-Take fast and slow paths to the same source vertex $x$; suppose two
+Take fast and slow paths to the same source vertex x; suppose two
angular effects on disjoint matched target patches have nonzero ordered
product. Choose smooth buffered angular cutoffs with bounded complementary
-thickness. Let $A$ be the fast cutoff and $L$ its one-ended limiting
-effect; let $B_1$ be the slow cutoff used on slow-link columns, with start
-patch disjoint from that of $A$. The finite corner-frame identity and
-exhaustion of fixed windows give bounded columns $f,f',b'$ for which the
-conjugate-linear functional, on $b(v)=sum_ih_i(v)b_i$ with smooth $h_i$
-and bounded columns $b_i$, is the slow limit of
+thickness. Let A be the fast cutoff and L its one-ended limiting
+effect; let B_1 be the slow cutoff used on slow-link columns, with start
+patch disjoint from that of A. The finite corner-frame identity and
+exhaustion of fixed windows give bounded columns f,f',b' for which the
+conjugate-linear functional, on b(v)=sum_ih_i(v)b_i with smooth h_i
+and bounded columns b_i, is the slow limit of
sum_iBigl< D_{h_i(q^+)B_1}[b'(ay)^*X_a(y)f'(y)],,
b_i(ay)^*X_a(y)(Lf)(y)Bigr>.
-Here $q^+$ is the image of the ordinary index $q$ (the reverse angular
+Here q^+ is the image of the ordinary index q (the reverse angular
variable in the matched component), and smooth boundary tests on the
large-gap windows extend through the simple top singular line in the
normalized proximal matrix chart.
-Bound it by the fast uncut form at $x$. At fixed slow time replace $Lf$
-by its defining fast weak limit along $t$; the result pairs
-$X_tX_a^*b_i(ay)$ with the transverse product of $D_AX_tf$ and the adjoint
-of the first column in (eq:input-4), with indices $p,q^{-1}$.
-Transfer $h_i$ by evaluation at the image of $qp^{-1}to q^+$. The
+Bound it by the fast uncut form at x. At fixed slow time replace Lf
+by its defining fast weak limit along t; the result pairs
+X_tX_a^*b_i(ay) with the transverse product of D_AX_tf and the adjoint
+of the first column in (eq:input-4), with indices p,q^{-1}.
+Transfer h_i by evaluation at the image of qp^{-1}to q^+. The
transfer error is uniform: normalized products are smooth with bounded
-derivatives on the buffered patches, so replacing $q$'s normalized matrix
+derivatives on the buffered patches, so replacing q's normalized matrix
by its rank-one top truncation writes the difference of the two angular
tests as a finite sum of entries of the rank-one defect times symbols with
uniformly bounded seminorms. The slow cut column multiplied by this defect
@@ -2955,32 +2954,32 @@ tends to zero in Hilbert norm; the small-factor assertion of
Proposition~prop:transfer-transverse kills the product error. The
remaining cut product has a uniform Hilbert norm. The squared norm of the
summed pairing column has fast limit equal to the fast uncut angular form
-on $sum_ih_iU_a^*b_i$, which by parabolic invariance at $a$ equals the
-form on $b$ at $x$. Cauchy--Schwarz gives the desired bound.
+on sum_ih_iU_a^*b_i, which by parabolic invariance at a equals the
+form on b at x. Cauchy--Schwarz gives the desired bound.
-Move the end columns by the reverse radical $n$ (invariance unchanged),
-giving a bound by the fast form on $sum_ih_iU_nb_i$, hence the form on $b$
-at $n^{-1}x$. Opposite radical moves cover an open Bruhat cell. Choose
+Move the end columns by the reverse radical n (invariance unchanged),
+giving a bound by the fast form on sum_ih_iU_nb_i, hence the form on b
+at n^{-1}x. Opposite radical moves cover an open Bruhat cell. Choose
two generic moved labels there; the same nonzero functional is bounded by
both forms, contradicting Lemma~lem:transfer-disjointness.
Thus candidate products at the same source vertex are supported on
-equality of target values. For unital POVMs $E,F$ with this property,
-regular approximation gives $E(S)F(T)=0$ for disjoint Borel $S,T$;
-unitality gives $E(S)=E(S)F(S)=F(S)$, so $E(S)^2=E(S)$ and the candidates
+equality of target values. For unital POVMs E,F with this property,
+regular approximation gives E(S)F(T)=0 for disjoint Borel S,T;
+unitality gives E(S)=E(S)F(S)=F(S), so E(S)^2=E(S) and the candidates
are equal PVMs.
## Incidence range containments
-Let $zsim F$ mean $z$ is incident to $F$. Fix a positive stratum of a
-mixed source face path or a pure vertex path $F$; denote its normalized
-image-face marginal by $Psi_F$ and the scalar mass of the unnormalized
-stratum by $c_F>0$. For an open patch $O$ of target type $p(i)$, put
-$mathcal I_i(O)=calI_i(O)={x:the target face x admits incident
-type-p(i) vertex in O}$. For an archimedean highest-line module
-$V$, let $R_x$ be the orthogonal projector onto
-$span{n(v):vsim x, type(v)=p(i)}$, and
+Let zsim F mean z is incident to F. Fix a positive stratum of a
+mixed source face path or a pure vertex path F; denote its normalized
+image-face marginal by Psi_F and the scalar mass of the unnormalized
+stratum by c_F>0. For an open patch O of target type p(i), put
+mathcal I_i(O)=calI_i(O)={x:the target face x admits incident
+type-p(i) vertex in O}. For an archimedean highest-line module
+V, let R_x be the orthogonal projector onto
+span{n(v):vsim x, type(v)=p(i)}, and
define
L_z=ranbarint n(v)n(v)^* ⊗ Phi_z(dv).
@@ -2991,90 +2990,90 @@ define
The following inclusions hold:
[label=(roman*)]
-item The support space of $Psi_F(mathcal I_i(O))$calI_i(O))$ is contained in
-$overline{sum_{zsim F}ranPhi_z(O)}$.
-item If the path is mixed and $p(i)$ is archimedean, in every even highest
-power the support of $int R_x ⊗ Psi_F(dx)$ is contained in
-$overline{sum_{zsim F}L_z}$.
+ The support space of Psi_F(mathcal I_i(O))calI_i(O)) is contained in
+overline{sum_{zsim F}ranPhi_z(O)}.
+ If the path is mixed and p(i) is archimedean, in every even highest
+power the support of int R_x ⊗ Psi_F(dx) is contained in
+overline{sum_{zsim F}L_z}.
For (i), a stronger individual-pair statement holds: on a compact
-full-data window that sends a buffered transverse end patch $V_1$ into
-$O$, if $finkerPhi_z(O)$ and $binranbarPhi_z(V_1)$, then
-$E(window)J(barbar b,f)=0$.b,f)=0$. On a mixed path $f$ may be any Hilbert
+full-data window that sends a buffered transverse end patch V_1 into
+O, if finkerPhi_z(O) and binranbarPhi_z(V_1), then
+E(window)J(barbar b,f)=0.b,f)=0. On a mixed path f may be any Hilbert
column; on a nonmixed path it is relatively bounded in the sense of
-Lemma~lem:links-relative-boundedness. In both cases $b$ may be any
+Lemma~lem:links-relative-boundedness. In both cases b may be any
vector in the stated closed end range.
Proof.
-We prove (i) by annihilating the common kernel $mathcal C=∩_calC=∩_{zsim
-F}kerPhi_z(O)$. For mixed paths take any $finmathcal C$calC$ and
-operator-bounded $f_lto f$. For nonmixed paths supported in $G_0$ with
-$N_0=mathcal N^calN^{G_0}$: if $i$ is in another component, ordinary fullness
-gives $mathcal C=0$calC=0$ and the inclusion is immediate; otherwise
-Lemma~lem:links-relative-boundedness(ii) makes $mathcal C$calC$
-$N_0$-invariant and part (iii) gives $f_lto f$ with
-$E_{N_0}(f_lf_l^*)le E_{N_0}(ff^*)$; in either case
-$< f_l,Phi_z(O)f_l>to0$ for each $zsim F$.
-
-Fix a compact window sending a buffered transverse $V_1$ into $O$,
-$zsim F$, and an operator-bounded $b_0$. A vertex path $s^{-1}$ toward
-$z$ with a smooth angular cut in $V_1$ and fixed complementary thickness
+We prove (i) by annihilating the common kernel mathcal C=∩_calC=∩_{zsim
+F}kerPhi_z(O). For mixed paths take any finmathcal CcalC and
+operator-bounded f_lto f. For nonmixed paths supported in G_0 with
+N_0=mathcal N^calN^{G_0}: if i is in another component, ordinary fullness
+gives mathcal C=0calC=0 and the inclusion is immediate; otherwise
+Lemma~lem:links-relative-boundedness(ii) makes mathcal CcalC
+N_0-invariant and part (iii) gives f_lto f with
+E_{N_0}(f_lf_l^*)le E_{N_0}(ff^*); in either case
+< f_l,Phi_z(O)f_l>to0 for each zsim F.
+
+Fix a compact window sending a buffered transverse V_1 into O,
+zsim F, and an operator-bounded b_0. A vertex path s^{-1} toward
+z with a smooth angular cut in V_1 and fixed complementary thickness
realizes its cut end effect by an auxiliary weak limit; exhausting cuts
-and varying $b_0$ gives a total set in
-$ranbarPhi_z(V_1)$. For fixed main time $a$ and index $l$,
-insert the auxiliary $s$-limit: cyclicity pairs
-$X_s(ay)X_a(y)f_l(y)$ with the product of cut $X_s(ay)b_0(ay)$ and
-$b''(ay)^*X_a(y)f''(y)$. If $q,h$ are the two product indices, the
-observed inverse is $h^{-1}q^{-1}$; transversality and the buffer put its
-image in $O$ once the auxiliary gap is large, so the pairing column has
-norm at most $< f_l,Phi_z(O)f_l>^{1/2}$. The main split
-translation fixes $z$ (trivially on Levi chamber refinements incident to
-$F$), so the bound is independent of main time. Proposition~prop:transfer-transverse
+and varying b_0 gives a total set in
+ranbarPhi_z(V_1). For fixed main time a and index l,
+insert the auxiliary s-limit: cyclicity pairs
+X_s(ay)X_a(y)f_l(y) with the product of cut X_s(ay)b_0(ay) and
+b''(ay)^*X_a(y)f''(y). If q,h are the two product indices, the
+observed inverse is h^{-1}q^{-1}; transversality and the buffer put its
+image in O once the auxiliary gap is large, so the pairing column has
+norm at most < f_l,Phi_z(O)f_l>^{1/2}. The main split
+translation fixes z (trivially on Levi chamber refinements incident to
+F), so the bound is independent of main time. Proposition~prop:transfer-transverse
gives a uniform Hilbert bound on the other cut product depending only on
-the fixed data, not on $|f_l|_∞$. Take the main-link limits with $l$
-fixed, then $lto∞$; on mixed paths the tensor isometry, on nonmixed
+the fixed data, not on |f_l|_∞. Take the main-link limits with l
+fixed, then lto∞; on mixed paths the tensor isometry, on nonmixed
paths the common conditional expectation bound, justifies the passage.
-Keep $f$ fixed, exhaust auxiliary cuts, and take the closed span of the
+Keep f fixed, exhaust auxiliary cuts, and take the closed span of the
resulting end ranges. Continuity in the entire Hilbert end variable
(mixed: tensor isometry; nonmixed: Lemma~lem:links-relative-boundedness)
-gives $E(window)J(barbar b,f)=0$b,f)=0$ for all
-$binranbarPhi_z(V_1)$.
+gives E(window)J(barbar b,f)=0b,f)=0 for all
+binranbarPhi_z(V_1).
Move the end occurrence also by the independent reverse radical. As
-$zsim F$ and these moves vary, the resulting labels cover a conull
+zsim F and these moves vary, the resulting labels cover a conull
transverse open cell; fullness and strong continuity make these end ranges
total, giving annihilation on the window for every end vector.
-For the frame step: exhaust the compact windows to the event $A$ of the
-chosen gap stratum and image condition $mathcal I_i(O)$;calI_i(O)$; monotone
-continuity gives $E(A)J(barbar b,f)=0$.b,f)=0$. Then $A$ is invariant under ignored
+For the frame step: exhaust the compact windows to the event A of the
+chosen gap stratum and image condition mathcal I_i(O);calI_i(O); monotone
+continuity gives E(A)J(barbar b,f)=0.b,f)=0. Then A is invariant under ignored
left Fourier shifts, and the frame identity
-$c_F< f,Psi_F(mathcal I_i(O))f>=calI_i(O))f>=frac1dsum_r
-|E(A)J(b_rbar,f)|^2=0$ (with $b_r$ the constant frame of $e$)
+c_F< f,Psi_F(mathcal I_i(O))f>=calI_i(O))f>=frac1dsum_r
+|E(A)J(b_rbar,f)|^2=0 (with b_r the constant frame of e)
combined with positivity and density gives (i). Arbitrary Borel output
subprojections are imposed after buffered tests.
-For (ii), write $S(h)$ for the Hilbert-normalized real module matrix at
-$h^{-1}$. If $(f_i)perp L_z$ and $(b_j)$ lies in the amplified line
+For (ii), write S(h) for the Hilbert-normalized real module matrix at
+h^{-1}. If (f_i)perp L_z and (b_j) lies in the amplified line
range on a patch transverse to the output reverse window, the same
computation gives
E(window)sum_{ij}E(S_{ij}(h))J(b_jbar,f_i)=0.
-The auxiliary fast symbol is $S(q)S(q)^*$ with cutoffs, and
-$S(h)S(q)S(q)^*=rho(h,q)S(qh)S(q)^*$ with $rho$ and derivatives bounded
-on transverse supports. Move $S(qh)^*$ to the full pairing column; its
+The auxiliary fast symbol is S(q)S(q)^* with cutoffs, and
+S(h)S(q)S(q)^*=rho(h,q)S(qh)S(q)^* with rho and derivatives bounded
+on transverse supports. Move S(qh)^* to the full pairing column; its
squared symbol tends in the auxiliary limit to the line-projector effect
-defining $L_z$, with the same stabilizer translation, so it vanishes on
+defining L_z, with the same stabilizer translation, so it vanishes on
the approximating kernel input. The reverse radical may again move the
end; amplified fullness makes the resulting end columns span the whole
-end space. The surviving top block of $S(h)$ has range containing every
+end space. The surviving top block of S(h) has range containing every
incident highest line (Lemma~lem:links-matrix-data); bounded local
right inverses or finite local spanning columns transfer the annihilation
-in (eq:input-5) to $R_x$. Compressing with mixed input independence
+in (eq:input-5) to R_x. Compressing with mixed input independence
from Proposition~prop:links-link proves (ii).
@@ -3089,7 +3088,7 @@ opposite vertices. Success is increasing in the initial Bruhat position.
If a word fails at a source pair, the ordered candidate product at that
pair is supported on target pairs for which the permuted word fails. The
sets of words successful at the least position are unchanged by the type
-permutation $p$.
+permutation p.
Proof.
@@ -3115,26 +3114,26 @@ If a word is successful at the least target position it is successful at
every target position. If the inverse-permuted word failed at a least
source pair, failure propagation would force the unital ordered source
product to have empty support, impossible. This gives inclusion of
-success languages in one direction; iterating $p$ (finite order) closes
+success languages in one direction; iterating p (finite order) closes
the cyclic chain.
[An inequality-detecting word]
-Every higher-rank component has a vertex type $j$ and an incidence word
-that fails exactly when two initial type-$j$ vertices are equal.
+Every higher-rank component has a vertex type j and an incidence word
+that fails exactly when two initial type-j vertices are equal.
Proof.
-For type $A$, take an end node and move the second vertex to an incident
+For type A, take an end node and move the second vertex to an incident
vertex of the opposite end type; success is equivalent to a hyperplane in
the second point's position avoiding the first. For other crystallographic
-diagrams, choose a self-opposite leaf $j$ with neighbor $k$ and round
-trips from $j$ to $k$ and back; each permits the block
-$W_{widehatwidehat j}j{1,s_j}W_{widehatwidehat j}j$. With $r$ the minimum number of
-$s_j$ factor in a reduced longest word, $r-1$ round trips cannot end
-opposite from equality, and $r$ can from the least nontrivial double
-coset $W_{widehatwidehat j}js_jW_{widehatwidehat j}j$. Every unequal position is above
+diagrams, choose a self-opposite leaf j with neighbor k and round
+trips from j to k and back; each permits the block
+W_{widehatwidehat j}j{1,s_j}W_{widehatwidehat j}j. With r the minimum number of
+s_j factor in a reduced longest word, r-1 round trips cannot end
+opposite from equality, and r can from the least nontrivial double
+coset W_{widehatwidehat j}js_jW_{widehatwidehat j}j. Every unequal position is above
that least nontrivial double coset, so success at all unequal pairs
follows from Lemma~lem:transfer-incidence-words.
@@ -3142,10 +3141,10 @@ follows from Lemma~lem:transfer-incidence-words.
[Sharp flag system]
All candidates are sharp and independent of candidate choice. For each
-source vertex $z$, face $F$, and chamber $C$ write the resulting PVMs as
-$P_z,P_F,P_C$. Incident source vertex measurements commute and are
+source vertex z, face F, and chamber C write the resulting PVMs as
+P_z,P_F,P_C. Incident source vertex measurements commute and are
supported on incidence of their target values. Every lexicographic or
-mixed face path to $F$ observes exactly the vertices of $p(F)$ with these
+mixed face path to F observes exactly the vertices of p(F) with these
measurements. Incident source labels commute across the two normal
measured copies.
@@ -3153,7 +3152,7 @@ measured copies.
Proof.
Lemma~lem:transfer-inequality-word and its permuted inverse give
failure of the corresponding word at equality of source
-$p^{-1}(j)$, so every ordered product of two candidates at that source
+p^{-1}(j), so every ordered product of two candidates at that source
label is supported on the target diagonal. The unital-POVM argument
(Proposition~prop:transfer-isolated-sharpness) makes them equal
projections, supplying a sharp type in every target component.
@@ -3175,19 +3174,19 @@ incidence family contains the maximal parabolic and is proper, so it is
exactly the vertex parabolic. Diagonal support gives sharpness and
uniqueness, component by component.
-For a general mixed face path of type $I$: the type-$i$ source vertex $z_i$
-incident to $F$ is unique for $iin I$. Proposition~prop:transfer-incidence(i)
-puts $Psi_F(mathcal I_i(O))$calI_i(O))$ inside $P_{z_i}(O)$. A target face omitting
-$p(i)$ admits two distinct incident vertices of that type, so disjoint
-$O,O'$ make $Psi_F(mathcal I_i(O)calI_i(O)capmathcal I_i(O'))$calI_i(O'))$ supported in
-both $P_{z_i}(O)$ and $P_{z_i}(O')$, hence zero. Countably many such
-pairs exhaust the omitted-type event, so every $p(i)$ for $iin I$
+For a general mixed face path of type I: the type-i source vertex z_i
+incident to F is unique for iin I. Proposition~prop:transfer-incidence(i)
+puts Psi_F(mathcal I_i(O))calI_i(O)) inside P_{z_i}(O). A target face omitting
+p(i) admits two distinct incident vertices of that type, so disjoint
+O,O' make Psi_F(mathcal I_i(O)calI_i(O)capmathcal I_i(O'))calI_i(O')) supported in
+both P_{z_i}(O) and P_{z_i}(O'), hence zero. Countably many such
+pairs exhaust the omitted-type event, so every p(i) for iin I
occurs. The upper bound of Proposition~prop:links-gap-count forces
-the stratum type to equal $p(I)$, with marginals $P_{z_i}$; sharp
+the stratum type to equal p(I), with marginals P_{z_i}; sharp
marginals put the joint measurement in their common multiplicative domain,
-giving $P_F$. Lexicographic path marginals are uniquely identified and
+giving P_F. Lexicographic path marginals are uniquely identified and
commute; no additional observed vertex type appears (an extra sharp
-marginal invariant under both the source face parabolic of $F$ and its
+marginal invariant under both the source face parabolic of F and its
own parabolic would by normalization give a source-invariant boundary
measurement, contradicting contraction). In the three-slot system, the
two normal copies are marginals of one common dilation and commute.
@@ -3195,7 +3194,7 @@ two normal copies are marginals of one common dilation and commute.
[Coxeter compatibility and ordered galleries]
-The permutation $p$ preserves the Coxeter matrix. Along a source chamber
+The permutation p preserves the Coxeter matrix. Along a source chamber
gallery, the ordered product of the chamber PVMs is supported on target
galleries of the permuted type, allowing stutters. Smooth functions
vanishing on that target gallery manifold annihilate the ordered product,
@@ -3216,9 +3215,9 @@ reduced longest source gallery: nonzero opposite endpoints require a
permuted-type gallery connecting opposite target chambers. Both ends use
the same full spherical Weyl system, so their longest elements have equal
Coxeter length, hence the permuted word is reduced longest and no stutter
-is possible. An alternating rank-two word of length $m_{ij}$ is reduced
+is possible. An alternating rank-two word of length m_{ij} is reduced
and extends to a reduced longest word; its permuted segment is reduced, so
-$m_{ij}le m_{p(i)p(j)}$; cycling the finite permutation gives equality.
+m_{ij}le m_{p(i)p(j)}; cycling the finite permutation gives equality.
## Exact residue links
@@ -3226,14 +3225,14 @@ $m_{ij}le m_{p(i)p(j)}$; cycling the finite permutation gives equality.
Take a mixed lex face path missing one type in a higher-rank component (or
one type in each of several). Fix one such component; its source rank-one
Levi boundary parametrizes matched opposite-panel completions at the two
-source ends, denoted $z$. Refining each of the two opposite source faces
+source ends, denoted z. Refining each of the two opposite source faces
by that same flag of their common Levi gives its start and ordinary-end
missing vertices, with the ordinary-end types relabeled by opposition.
-Write $P_z,Q_z$ for the sharp missing-vertex PVMs at these vertices. Their
-common source parameter transports by $zmapsto sz$ under simultaneous
+Write P_z,Q_z for the sharp missing-vertex PVMs at these vertices. Their
+common source parameter transports by zmapsto sz under simultaneous
Levi action. Frame the two sharp endpoint faces separately by compact
transports, identify the reference opposite panels by transverse
-projection; the residual output coordinate $g$ is then a rank-one target
+projection; the residual output coordinate g is then a rank-one target
projectivity from the end pencil to the start pencil.
[Residue graph equation]
@@ -3254,48 +3253,48 @@ exact input translations may be made before any further limit.
Proof.
Work in an archimedean component. On the ordinary-end Hilbert space use
-the slots in order: reverse panel, end missing vertex at $z$, observation
+the slots in order: reverse panel, end missing vertex at z, observation
at the start label (possibly moved by the reverse radical), with value
-$v$ in the last slot. A source gallery shortened from the longest by the
+v in the last slot. A source gallery shortened from the longest by the
missing reflection connects the matched chambers; its target gallery has
the same shortened type (Proposition~prop:transfer-galleries). The
-shortened Weyl position $s_iw_0=w_0s_{i^*}$ has all other descents, so
-parabolic saturation cannot reach $w_0$; the missing vertices are
+shortened Weyl position s_iw_0=w_0s_{i^*} has all other descents, so
+parabolic saturation cannot reach w_0; the missing vertices are
nonopposite. A vertex transverse to a panel of dual missing type has
exactly one incident completion whose missing vertex is nonopposite (the
ordinary rank-one parabolic projection), so the ordered gallery equation
-says the end missing vertex is $pr(v)$ on a transverse
+says the end missing vertex is pr(v) on a transverse
window; integrating intermediate slots by identity projections preserves
the equation.
-Use (eq:input-5) at $z$, moving only its end occurrence by the
+Use (eq:input-5) at z, moving only its end occurrence by the
reverse radical. Sharpness permits the start column to be
-$(1-n(P_z)n(P_z)^*)$ applied to an arbitrary column and the end column to
+(1-n(P_z)n(P_z)^*) applied to an arbitrary column and the end column to
be the moved line projector; the link is annihilated by
-$(1-n(u)n(u)^*)S(h)n(v)$. The two angular inputs commute (separate
+(1-n(u)n(u)^*)S(h)n(v). The two angular inputs commute (separate
tensor slots); separation extends to arbitrary smooth joint angular
multipliers. On the stratum the matrix action is the residual projectivity
after transverse projection, so the equations are the line graph
-$u=gpr(v)$, which generates its smooth vanishing ideal on
+u=gpr(v), which generates its smooth vanishing ideal on
compact transverse windows (the highest-line embedding is immersive).
Output coordinates may be Borel parameters with locally uniform angular
derivative bounds.
Combine the two steps in order. Preapply at the end a reverse-panel cut
followed by a moved-vertex cut on a buffered transverse rectangle; leave
-the start vector arbitrary. Insert the $Q_z$ slit next to the reverse
+the start vector arbitrary. Insert the Q_z slit next to the reverse
panel (they commute); the shortened gallery replaces its value by
-$pr(v)$. Integrate out that slot; pull the sharp
-reverse-panel projection through $J$ (Proposition~prop:links-link);
-the line equation gives $u=gpr(v)=gq$. This proves
+pr(v). Integrate out that slot; pull the sharp
+reverse-panel projection through J (Proposition~prop:links-link);
+the line equation gives u=gpr(v)=gq. This proves
(eq:input-6) on those ordered ranges; no discontinuous joint symbol
has been applied to noncommuting end measurements.
The ordered ranges are total: their closed join is target invariant, fixed
by a split source subgroup escaping in every ambient component, so scalar
-mass (Proposition~prop:links-link). Varying $z$ gives nonzero ranges
+mass (Proposition~prop:links-link). Varying z gives nonzero ranges
(visible start label, patch fullness); Levi transitivity gives nonzeroness
-for every $z$. Hence the join is the whole end space.
+for every z. Hence the join is the whole end space.
For a finite target place, replace the Hilbert line argument by the
individual-pair assertion of Proposition~prop:transfer-incidence on
@@ -3320,7 +3319,7 @@ amplifications if necessary. For product-almost every pair of endpoint
fibres, the mixed isometry (eq:input-1), the graph equation
(eq:input-6), simultaneous source Levi invariance, and exact input
translations hold. On almost every individual fibre, for every even
-highest power in an archimedean target module, the line ranges over $z$
+highest power in an archimedean target module, the line ranges over z
span at least the Hilbert fibre tensored with the span of all ordinary
incident highest lines. Consequently a homogeneous polynomial relation
with linear Hilbert-vector coefficients that holds spectrally at every
@@ -3355,78 +3354,78 @@ all ordinary incident highest lines. A polynomial vanishing on spectral
line ranges is orthogonal to their span, hence to each ordinary incident
highest line; norm powers homogenize different even degrees.
-For a relation known only on a chart ${Deltane0}$, clear a common
-homogeneous denominator and call the numerator $F_num$; global
-polynomial fullness applied to $Delta F_num$ vanishes spectrally
-on the chart by the relation and on ${Delta=0}$ by the factor; divide
-by $Delta$ on the chart. Pluecker equations and common local
+For a relation known only on a chart {Deltane0}, clear a common
+homogeneous denominator and call the numerator F_num; global
+polynomial fullness applied to Delta F_num vanishes spectrally
+on the chart by the relation and on {Delta=0} by the factor; divide
+by Delta on the chart. Pluecker equations and common local
denominators place algebraic membership tests in this form.
# Finite-place panels: fixed algebras and uniform upper speed
Take a mixed lexicographic face path missing one type in a non-isolated
-component, or one type in each of several. Write $B_S,B_T$ for the source
-and target panel boundaries. Let $S$ be the effective root-generated
-source panel group and $T$ the effective target adjoint point group, both
+component, or one type in each of several. Write B_S,B_T for the source
+and target panel boundaries. Let S be the effective root-generated
+source panel group and T the effective target adjoint point group, both
over finite extensions of a prime local field. The prime-field Lie algebra
-of $T$ is simple, $S$ is simple after removal of its ineffective center,
-and $T$ embeds faithfully in the automorphism group of its prime-field Lie
-algebra, so the centralizer in $T$ of an open subgroup of its
+of T is simple, S is simple after removal of its ineffective center,
+and T embeds faithfully in the automorphism group of its prime-field Lie
+algebra, so the centralizer in T of an open subgroup of its
root-generated group is trivial.
-There is an actual source Levi cover $widetilde Sto S$ acting strongly
+There is an actual source Levi cover widetilde Sto S acting strongly
continuously on the endpoint spaces, commuting with the sharp
-endpoint-face observations, moving residue labels by $zmapsto sz$, and
+endpoint-face observations, moving residue labels by zmapsto sz, and
its ineffective finite center acts trivially on the residue algebras. Our
-goal is a fixed homogeneous analytic homeomorphism $tau:B_Sto B_T$ and
-measurable target projectivity observations $a,b$ so that the endpoint
-families are evaluations of $zmapsto atau(z)$ and $zmapsto btau(z)$.
+goal is a fixed homogeneous analytic homeomorphism tau:B_Sto B_T and
+measurable target projectivity observations a,b so that the endpoint
+families are evaluations of zmapsto atau(z) and zmapsto btau(z).
## The paired endpoint algebra
-For a locally constant scalar $f$ on $B_T$, define the paired
-decomposable operator on $L^2(T;H') ⊕ L^2(T;I)$ by
+For a locally constant scalar f on B_T, define the paired
+decomposable operator on L^2(T;H') ⊕ L^2(T;I) by
- F_{z,f}(t)=(f(tP_z),f(tQ_z)),qquad tin T,
+ F_{z,f}(t)=(f(tP_z),f(tQ_z)), tin T,
-with spectral evaluation $f(tP_z)=int f(tu)P_z(du)$ and similarly at
-the end. Let $D_*$ be the von Neumann algebra generated by these over a
-countable dense set of labels and clopen tests; $D_*$ has separable
+with spectral evaluation f(tP_z)=int f(tu)P_z(du) and similarly at
+the end. Let D_* be the von Neumann algebra generated by these over a
+countable dense set of labels and clopen tests; D_* has separable
predual.
-In the intended model $f$ would be $f(tatau(z))$, with frame/projectivity
-entering only via $x=ta$. The result below says the actual paired
-algebra is exactly $L^∞(T)$ with left translation, establishing
+In the intended model f would be f(tatau(z)), with frame/projectivity
+entering only via x=ta. The result below says the actual paired
+algebra is exactly L^∞(T) with left translation, establishing
commutation of the endpoint families and the homogeneous form.
[The fixed algebra]
-For a sufficiently small compact-open $U<T$, the fixed algebra
-$D_*^U$ is atomic (a countable product of finite matrix algebras), and
-$D_*^T=mathbb C1$.bbC1$.
+For a sufficiently small compact-open U<T, the fixed algebra
+D_*^U is atomic (a countable product of finite matrix algebras), and
+D_*^T=mathbb C1.bbC1.
Proof.
-A $U$-fixed decomposable operator is essentially constant on left
-cosets, so there are normal evaluations $pi_t^+(d)=d_t^+$,
-$pi_r^-(d)=d_r^-$ for $Ut,Urin Ubackslash T$, jointly faithful on the
-countable set $Ubackslash T$. Identify
-$overlineoverline II ⊗ H'congHS(I,H')$ via
-$overlinexi ⊗ etamapsto|eta><xi|$. For fixed
-cosets $Ut,Ur$, put $Ω_{t,r}={g:Utg=Ur}$ and
-$K=J^*E(Ω_{t,r})zetainHS(I,H')$. Then for every
-$din D_*^U$,
+A U-fixed decomposable operator is essentially constant on left
+cosets, so there are normal evaluations pi_t^+(d)=d_t^+,
+pi_r^-(d)=d_r^- for Ut,Urin Ubackslash T, jointly faithful on the
+countable set Ubackslash T. Identify
+overlineoverline II ⊗ H'congHS(I,H') via
+overlinexi ⊗ etamapsto|eta><xi|. For fixed
+cosets Ut,Ur, put Ω_{t,r}={g:Utg=Ur} and
+K=J^*E(Ω_{t,r})zetainHS(I,H'). Then for every
+din D_*^U,
d_t^+K=Kd_r^-.
-Varying $zeta$, these intertwiners force the fibres $d_t^+$ to be graded
+Varying zeta, these intertwiners force the fibres d_t^+ to be graded
by the normal representation types: each Schur subspace remains
finite-dimensional per coset, giving a countable direct sum of finite
-matrix algebras. Ergodicity of $T$ acting transitively on $Ubackslash
-T$ forces $D_*^T=mathbb C1$:bbC1$: any $T$-fixed normal functional on the
+matrix algebras. Ergodicity of T acting transitively on Ubackslash
+T forces D_*^T=mathbb C1:bbC1: any T-fixed normal functional on the
countable Cartesian product vanishes off the identity coset, and the
resulting character is scalar.
@@ -3447,10 +3446,10 @@ a projective unitary group. A noncompact simple real Lie algebra has no
nonzero homomorphism into a compact Lie algebra (simplicity would make it
injective, and a positive definite invariant inner product would make
the source compact); a homomorphism with zero differential from a
-connected Lie group is trivial. Thus the source would fix every $D_*^U$
-pointwise. As compact open $U$ decreases to the identity, the normal
-averages $int_Ualpha_u(d)du$ converge ultraweakly to $d$, so the ranges
-generate $D_*$; contradiction to Proposition~prop:finite-frame.
+connected Lie group is trivial. Thus the source would fix every D_*^U
+pointwise. As compact open U decreases to the identity, the normal
+averages int_Ualpha_u(d)du converge ultraweakly to d, so the ranges
+generate D_*; contradiction to Proposition~prop:finite-frame.
Finiteness and bijectivity of the type permutation give the symmetric
assertion.
@@ -3458,225 +3457,225 @@ assertion.
## Corner reduction and finite blocks
Both panel groups are now nonarchimedean. Denote the residue
-characteristics by $p_S,p_T$. Choose small torsion-free logarithmic
-compact open subgroups $V<S$ and $U<T$, respectively pro-$p_S$ and
-pro-$p_T$, in faithful adjoint matrix coordinates. Choose a minimal
-projection $ein D_*^U$; shrink $V$ to fix $e$ (an atomic central block
+characteristics by p_S,p_T. Choose small torsion-free logarithmic
+compact open subgroups V<S and U<T, respectively pro-p_S and
+pro-p_T, in faithful adjoint matrix coordinates. Choose a minimal
+projection ein D_*^U; shrink V to fix e (an atomic central block
has open stabilizer by the normal-state test). The image of that compact
profinite stabilizer in the automorphism group of its finite matrix block
is profinite and Chinese-Lie-Lie closed, hence finite; take a smaller
-$V$ inside its open kernel. Put
+V inside its open kernel. Put
- D_0=eD_*e,qquad D_0^U=mathbb Ce.bbCe.
+ D_0=eD_*e, D_0^U=mathbb Ce.bbCe.
-Choose decreasing compact open normal subgroups $U_n△left U$ with
-intersection ${1}$; then $D_n=D_0^{U_n}$ is finite-dimensional, since
-an orthogonal family of nonzero projections has at most $[U:U_n]$ members
-(their average mass forces $lambda_age[U:U_n]^{-1}$).
+Choose decreasing compact open normal subgroups U_n△left U with
+intersection {1}; then D_n=D_0^{U_n} is finite-dimensional, since
+an orthogonal family of nonzero projections has at most [U:U_n] members
+(their average mass forces lambda_age[U:U_n]^{-1}).
[Block stabilizers]
-The simple blocks of $D_n$ form a transitive $U$-set; the stabilizer
-$L_n<U$ of a chosen block acts on it by an irreducible projective unitary
-representation. If $G_n<Vtimes U$ is the stabilizer of that block for
-the commuting source and target actions, then $G_n$ is closed, its
-projection onto $V$ is surjective, and
+The simple blocks of D_n form a transitive U-set; the stabilizer
+L_n<U of a chosen block acts on it by an irreducible projective unitary
+representation. If G_n<Vtimes U is the stabilizer of that block for
+the commuting source and target actions, then G_n is closed, its
+projection onto V is surjective, and
- [K_n,K_n] acts trivially on D_n,qquad
+ [K_n,K_n] acts trivially on D_n,
K_n={vin V:(v,1)in G_n}.
Proof.
-Ergodicity of $U/U_n$ on $D_n$ makes its center $U$-transitive. The
-source $V$ commutes with $D_*$ and acts by $sigma$-conjugation, so it
-normalizes $G_n$. For each $vin V$, surjectivity onto $V$ follows by
-choosing a block $b$, then a conjugate $sigma_v(b)$ at the vacuous level.
-Commutators $[K_n,K_n]$ commute both with the action and with $v$; the
+Ergodicity of U/U_n on D_n makes its center U-transitive. The
+source V commutes with D_* and acts by sigma-conjugation, so it
+normalizes G_n. For each vin V, surjectivity onto V follows by
+choosing a block b, then a conjugate sigma_v(b) at the vacuous level.
+Commutators [K_n,K_n] commute both with the action and with v; the
source and target normal subgroups meet in the identity because the
-target $U$ is torsion-free, so their nonparallel action kills the block.
+target U is torsion-free, so their nonparallel action kills the block.
-Let $H$ be any closed subgroup of the fixed product $Vtimes U$ with
+Let H be any closed subgroup of the fixed product Vtimes U with
fixed faithful multiplicative coordinates. The following uniform lemma
controls its logarithms.
[Uniform logarithm saturation]
-There is a constant $c$, depending only on $Vtimes U$, such that for
-every closed subgroup $Hsubset Vtimes U$,
+There is a constant c, depending only on Vtimes U, such that for
+every closed subgroup Hsubset Vtimes U,
exp(p^c(Lie Hcap M_H))subset H,
-where $M_H=log H$ in the chosen coordinates and the image is a fixed
+where M_H=log H in the chosen coordinates and the image is a fixed
small ball.
Proof.
Use the Jordan--Holder filtration of the prime-field Lie algebra in the
-given coordinates, and write $exp(x)$ for the resulting truncated series
-on $p^c$. By Cayley--Hamilton on the integer action lattice, the relevant
-partial matrices stay integral, so every element of $p^c(Lie Hcap M_H)$
+given coordinates, and write exp(x) for the resulting truncated series
+on p^c. By Cayley--Hamilton on the integer action lattice, the relevant
+partial matrices stay integral, so every element of p^c(Lie Hcap M_H)
arises from a closed-subgroup element up to a bounded shift, saturated by
-the standard BCH/Dynkin estimates at the fine $p$-adic scale. Details are
+the standard BCH/Dynkin estimates at the fine p-adic scale. Details are
as in the original.
[Limiting source kernel]
-The source $V$-action on $D_0$ has zero Lie-algebra kernel: for every
-open subgroup $V_1<V$ acting trivially on $D_0$, we have $V_1={1}$.
+The source V-action on D_0 has zero Lie-algebra kernel: for every
+open subgroup V_1<V acting trivially on D_0, we have V_1={1}.
Proof.
-As $nto∞$, $D_n$ exhausts $D_0$ ultraweakly (normal averaging and
-strong continuity). Since $V$ is pro-$p$, any nontrivial open kernel
+As nto∞, D_n exhausts D_0 ultraweakly (normal averaging and
+strong continuity). Since V is pro-p, any nontrivial open kernel
contains a commutator subgroup of an open normal subgroup, and we may
-reduce to a case where a fixed open $V_1$ acts trivially on all $D_n$.
-Choose small logarithmic $V_1subset V$ and $U$-compatible $D_n$. Because
-$G_n$ projects onto $V$, every block's stabilizer contains an analytic
-subgroup; a limiting source-only kernel would force one fixed open $V_1$
-into all late $G_n$'s. Then $[V_1,V_1]$ fixes $D_n$ at every sufficiently
-late level, so $[V_1,V_1]subsetLie V_1$ satisfies
-$[mathfrak v_1,frakv_1,mathfrak v_1]=frakv_1]=mathfrak v_1$frakv_1$ and hence $V_1$ is
+reduce to a case where a fixed open V_1 acts trivially on all D_n.
+Choose small logarithmic V_1subset V and U-compatible D_n. Because
+G_n projects onto V, every block's stabilizer contains an analytic
+subgroup; a limiting source-only kernel would force one fixed open V_1
+into all late G_n's. Then [V_1,V_1] fixes D_n at every sufficiently
+late level, so [V_1,V_1]subsetLie V_1 satisfies
+[mathfrak v_1,frakv_1,mathfrak v_1]=frakv_1]=mathfrak v_1frakv_1 and hence V_1 is
nontrivial, in contradiction to the hypothesis.
## Hausdorff limits and dimension equality
-The compact metrizable $Vtimes U$ has a compact space of closed
-subgroups in the Hausdorff topology. Every limit $G_∞$ of the $G_n$
-still projects onto $V$ (for $vin V$, choose $(v,u_n)in G_n$ and pass
-to a convergent subsequence of $u_n$). A nonzero source-only kernel at
-level $G_∞$ would force an open source subgroup into all
-sufficiently late $G_n$, contradicting Lemma~lem:finite-limit-kernel.
+The compact metrizable Vtimes U has a compact space of closed
+subgroups in the Hausdorff topology. Every limit G_∞ of the G_n
+still projects onto V (for vin V, choose (v,u_n)in G_n and pass
+to a convergent subsequence of u_n). A nonzero source-only kernel at
+level G_∞ would force an open source subgroup into all
+sufficiently late G_n, contradicting Lemma~lem:finite-limit-kernel.
Hence:
[Dimensions and abelian corner]
-$dim_{mathbb Q_p}bbQ_pmathfrak s=fraks=dim_{mathbb Q_p}bbQ_pmathfrak t$.frakt$. Every
-Hausdorff limit $G_∞$ has both projections isomorphisms, and the
-target-only subgroup $G_∞cap(1times U)$ is ${1}$. Consequently
-$D_0$ is abelian and equals $L^∞(U)$ with Haar probability.
+dim_{mathbb Q_p}bbQ_pmathfrak s=fraks=dim_{mathbb Q_p}bbQ_pmathfrak t.frakt. Every
+Hausdorff limit G_∞ has both projections isomorphisms, and the
+target-only subgroup G_∞cap(1times U) is {1}. Consequently
+D_0 is abelian and equals L^∞(U) with Haar probability.
Proof.
-Choose a basis of a fixed lattice $M_V$ and concat the matrices
-$B_{i,n}$; a nonzero full-rank square minor in the limiting
+Choose a basis of a fixed lattice M_V and concat the matrices
+B_{i,n}; a nonzero full-rank square minor in the limiting
concatenation, adjugated with fixed determinant valuations, gives
-$p^aM_Vtimes{0}subset(sum_iB_{i,n}M_V)times{0}subset M_n$
+p^aM_Vtimes{0}subset(sum_iB_{i,n}M_V)times{0}subset M_n
uniformly at late levels. By Lemma~lem:finite-uniform-log in the
-product logarithm group, $exp(p^{c+a}M_V)times1subset G_n$ uniformly.
-After enlarging $a$, $exp(p^{c+a}M_V)$ is a fixed open $V_1subset V$
-whose commutator $[V_1,V_1]$ fixes $D_n$ for all sufficiently late $n$
-in the subsequence; as $D_n$ exhausts $D_0$, $[V_1,V_1]$ fixes $D_0$,
-contradicting Lemma~lem:finite-limit-kernel. So $Lie(G_∞)$'s
+product logarithm group, exp(p^{c+a}M_V)times1subset G_n uniformly.
+After enlarging a, exp(p^{c+a}M_V) is a fixed open V_1subset V
+whose commutator [V_1,V_1] fixes D_n for all sufficiently late n
+in the subsequence; as D_n exhausts D_0, [V_1,V_1] fixes D_0,
+contradicting Lemma~lem:finite-limit-kernel. So Lie(G_∞)'s
source-only part has zero Lie algebra. Both projections of
-$Lie(G_∞)$ are then isomorphisms (surjections with equal dimension
-$dimmathfrak s=fraks=dimmathfrak t$).frakt$). The target-only subgroup is a compact
-zero-dimensional analytic subgroup of the torsion-free $U$, hence
-${1}$.
+Lie(G_∞) are then isomorphisms (surjections with equal dimension
+dimmathfrak s=fraks=dimmathfrak t).frakt). The target-only subgroup is a compact
+zero-dimensional analytic subgroup of the torsion-free U, hence
+{1}.
-Now fix $m$; for large $n$, every block stabilizer lies in $U_m$ and
-fixes the image of $D_m$ in the block pointwise. Its irreducible
-representation is then scalar; blocks are jointly faithful, so $D_m$ is
-abelian, and $D_0$ is abelian by exhaustion. The spectra $X_n$ of $D_n$
-are finite transitive $U$-spaces with surjective equivariant maps
-$X_{n+1}to X_n$; averaging a faithful normal state gives invariant
-uniform measures, so $D_0=L^∞(X,nu)$ with $X=varprojlim X_n$. The
-$U$-action on $X$ is transitive (nested chain of cosets), and uniform
-shrinking of stabilizers makes any point stabilizer trivial, so $X$ is
-the translation $U$-torsor with Haar probability.
+Now fix m; for large n, every block stabilizer lies in U_m and
+fixes the image of D_m in the block pointwise. Its irreducible
+representation is then scalar; blocks are jointly faithful, so D_m is
+abelian, and D_0 is abelian by exhaustion. The spectra X_n of D_n
+are finite transitive U-spaces with surjective equivariant maps
+X_{n+1}to X_n; averaging a faithful normal state gives invariant
+uniform measures, so D_0=L^∞(X,nu) with X=varprojlim X_n. The
+U-action on X is transitive (nested chain of cosets), and uniform
+shrinking of stabilizers makes any point stabilizer trivial, so X is
+the translation U-torsor with Haar probability.
-## The $T$-torsor coordinate
+## The T-torsor coordinate
[The full frame algebra]
-There is a $T$-equivariant normal isomorphism $D_*cong L^∞(T)$ with
+There is a T-equivariant normal isomorphism D_*cong L^∞(T) with
left translation. In these coordinates, the commuting source action is
- (beta_s f)(x)=f(xeta(s)),qquad eta:Sto T,
+ (beta_s f)(x)=f(xeta(s)), eta:Sto T,
-where $eta$ is a continuous homomorphism with isomorphic
-$mathbb Q_p$-differential.bbQ_p$-differential.
+where eta is a continuous homomorphism with isomorphic
+mathbb Q_p-differential.bbQ_p-differential.
Proof.
-By Lemma~prop:finite-dimensions, $e$ is an abelian projection. On
-its central support $z_e$ the algebra is type I, and
+By Lemma~prop:finite-dimensions, e is an abelian projection. On
+its central support z_e the algebra is type I, and
- Z(D_*)z_e→ eD_*e,qquad a⟼ ae
+ Z(D_*)z_e→ eD_*e, a⟼ ae
is a normal isomorphism. In the type I decomposition of this central
-summand, $e$ has rank one in every factor, and full central support makes
+summand, e has rank one in every factor, and full central support makes
the scalar field faithful.
-Countably many $T$-translates of $e$ have join one, so $D_*$ is type I.
-Its center has a positive-measure $U$-invariant chart isomorphic to $U$,
-and the $T$-action is ergodic (Equation~(eq:finite-fixed)).
-Use Mackey's Borel point realization [cite]: let $X$ realize
-the center, let $varphi:Uto X$ be measurable onto the chart.
-Fubini in the two Haar variables gives $x_0$ with
-$varphi(u)=ux_0$ almost everywhere; for a generic $u_0$,
-$varphi(vu_0)=vvarphi(u_0)$ for almost every $v$. Let
-$L=Stab_T(x_0)$. If $hin Lcap U$, comparing $u$ and
-$uh$ on a common conull set forces $h=1$, so $Lcap U$ meets only the
-identity in a small neighborhood, hence $L$ is discrete and closed. The
-orbit map $T/Lto X$ is a Borel bijection with Borel inverse, countably
+Countably many T-translates of e have join one, so D_* is type I.
+Its center has a positive-measure U-invariant chart isomorphic to U,
+and the T-action is ergodic (Equation~(eq:finite-fixed)).
+Use Mackey's Borel point realization [cite]: let X realize
+the center, let varphi:Uto X be measurable onto the chart.
+Fubini in the two Haar variables gives x_0 with
+varphi(u)=ux_0 almost everywhere; for a generic u_0,
+varphi(vu_0)=vvarphi(u_0) for almost every v. Let
+L=Stab_T(x_0). If hin Lcap U, comparing u and
+uh on a common conull set forces h=1, so Lcap U meets only the
+identity in a small neighborhood, hence L is discrete and closed. The
+orbit map T/Lto X is a Borel bijection with Borel inverse, countably
many translates of the chart cover modulo null sets, and
-$Z(D_*)cong L^∞(T/L)$.
+Z(D_*)cong L^∞(T/L).
-The measurable automorphisms of $T/L$ commuting with left translation
-are the right translations by $N_T(L)/L$ (explicitly: for such $F$,
-$gmapsto g^{-1}F(gL)$ is left-invariant, hence constant $=nL$, giving
-$n^{-1}Lnsubset L$, equality by applying to $F^{-1}$). The injection
-$N_T(L)/LtoAut(T/L)$ has a Borel inverse. The source
+The measurable automorphisms of T/L commuting with left translation
+are the right translations by N_T(L)/L (explicitly: for such F,
+gmapsto g^{-1}F(gL) is left-invariant, hence constant =nL, giving
+n^{-1}Lnsubset L, equality by applying to F^{-1}). The injection
+N_T(L)/LtoAut(T/L) has a Borel inverse. The source
action gives a measurable (hence continuous, analytic over
-$mathbb Q_p$)bbQ_p$) homomorphism $eta:Sto N_T(L)/L$.
-
-Its differential is injective: otherwise a small kernel $V_1<V$ acts
-trivially on the center; since $V$ fixes $e$, Equation~(eq:finite-center-corner)
-forces $V_1$ to act trivially on $eD_*e$, contradicting
-Lemma~lem:finite-limit-kernel. Since $L$ is discrete and
-$dimmathfrak s=fraks=dimmathfrak t$,frakt$, the closed Lie subgroup $N_T(L)$ has
-dimension equal to $T$ and is open. For $lin L$, conjugation from the
-open $N_T(L)$ into the discrete $L$ is continuous, so $l$ centralizes an
-open subgroup; the open-centralizer assertion forces $l=1$. Thus
-$L={1}$ and $Z(D_*)=L^∞(T)$ with left translation.
+mathbb Q_p)bbQ_p) homomorphism eta:Sto N_T(L)/L.
+
+Its differential is injective: otherwise a small kernel V_1<V acts
+trivially on the center; since V fixes e, Equation~(eq:finite-center-corner)
+forces V_1 to act trivially on eD_*e, contradicting
+Lemma~lem:finite-limit-kernel. Since L is discrete and
+dimmathfrak s=fraks=dimmathfrak t,frakt, the closed Lie subgroup N_T(L) has
+dimension equal to T and is open. For lin L, conjugation from the
+open N_T(L) into the discrete L is continuous, so l centralizes an
+open subgroup; the open-centralizer assertion forces l=1. Thus
+L={1} and Z(D_*)=L^∞(T) with left translation.
Remove matrix fibres: the type I decomposition and ergodicity of the
-center give $D_*cong L^∞(T;B(mathcal V))$calV))$ with constant separable
+center give D_*cong L^∞(T;B(mathcal V))calV)) with constant separable
dimension. In a measurable trivialization the action is
-$(alpha_ad)(x)=c(a,x)d(a^{-1}x)$ with
-$c(ab,x)=c(a,x)c(b,a^{-1}x)$ (an $Aut(B(mathcal V))$-calV))$-
+(alpha_ad)(x)=c(a,x)d(a^{-1}x) with
+c(ab,x)=c(a,x)c(b,a^{-1}x) (an Aut(B(mathcal V))-calV))-
valued cocycle). Changing variables and applying Fubini, freeze a generic
-$x_0$ and define $b(x)=c(xx_0^{-1},x)$; then $c(a,x)=b(x)b(a^{-1}x)^{-1}$
-almost everywhere, and changing the fibre trivialization by $b$ removes
+x_0 and define b(x)=c(xx_0^{-1},x); then c(a,x)=b(x)b(a^{-1}x)^{-1}
+almost everywhere, and changing the fibre trivialization by b removes
the cocycle. Pure translation fixes the constant field
-$B(mathcal V)$;calV)$; Equation~(eq:finite-fixed) forces $dimmathcal
-V=1$.calV=1$.
+B(mathcal V);calV); Equation~(eq:finite-fixed) forces dimmathcal
+V=1.calV=1.
## Ordinary homogeneous panel maps
[Recovery of the observed panels]
-The families $(P_z)$ and $(Q_z)$ commute on their respective start and
+The families (P_z) and (Q_z) commute on their respective start and
end spaces, and there are a homogeneous analytic homeomorphism
-$tau:B_Sto B_T$ and measurable target projectivity observations
-$a,b$ such that these families are evaluations of
+tau:B_Sto B_T and measurable target projectivity observations
+a,b such that these families are evaluations of
- z⟼ atau(z),qquad z⟼ btau(z).
+ z⟼ atau(z), z⟼ btau(z).
On the independent input tensor product, the output projectivity is
-sharp and equals $ab^{-1}$:
+sharp and equals ab^{-1}:
E(W)J=J,1_W(ab^{-1})quadfor every Borel Wsubset T.
@@ -3685,61 +3684,61 @@ All statements hold coordinatewise on many-panel links.
Proof.
-Commutativity of $D_*$ makes the paired generators' commutators vanish
+Commutativity of D_* makes the paired generators' commutators vanish
at almost every external frame; use a common conull set for countably
-many labels and clopen tests. The transformed tests generate $C(B_T)$,
+many labels and clopen tests. The transformed tests generate C(B_T),
so the original families commute, and label continuity extends to all
fixed pairs.
-In the intrinsic $D_*=L^∞(T)$ of
+In the intrinsic D_*=L^∞(T) of
Proposition~prop:finite-torsor, the spectral values give a
-jointly measurable boundary map $F(x,z)$ (countable determining
-observations plus label continuity). Frame covariance $F(ax,z)=aF(x,z)$
-and Fubini give $F(x,z)=xF_0(z)$, and the source action gives
+jointly measurable boundary map F(x,z) (countable determining
+observations plus label continuity). Frame covariance F(ax,z)=aF(x,z)
+and Fubini give F(x,z)=xF_0(z), and the source action gives
F_0(sz)=eta(s)F_0(z)
-in the ordinary measure class on $B_S$.
+in the ordinary measure class on B_S.
Regard each effective adjoint group over its coefficient field as a
-restriction of scalars to $mathbb Q_p$;bbQ_p$; in characteristic zero, all
+restriction of scalars to mathbb Q_p;bbQ_p; in characteristic zero, all
derivations of a semisimple Lie algebra are inner, and the identity
component of its algebraic automorphism group is the adjoint algebraic
-group. Conjugation by $A=deta$ is an isomorphism of these adjoint
-algebraic groups over $mathbb Q_p$,bbQ_p$, and
+group. Conjugation by A=deta is an isomorphism of these adjoint
+algebraic groups over mathbb Q_p,bbQ_p, and
- AAd(s)=Ad(eta(s))A,qquad sin S.
+ AAd(s)=Ad(eta(s))A, sin S.
Minimal rational parabolics are preserved under algebraic group
isomorphisms, and those of the restriction of scalars are exactly the
restrictions of the original rational parabolics (after splitting,
the descent condition identifies with a single parabolic over the
-original field); thus $tau:B_Sto B_T$ exists, intertwining $S$ with
-$eta(S)$.
-
-Write $B_S=S/P$; the measurable lift $smapstoeta(s)^{-1}F_0(sP)$ is
-essentially left invariant, hence constant; right-$P$ consistency makes
-it fixed by $eta(P)$, and a minimal parabolic fixes exactly its point on
-a rank-one boundary, so the constant is $tau$ and $F_0=tau$ almost
-everywhere. Both sides are continuous in measure in $z$, so the
+original field); thus tau:B_Sto B_T exists, intertwining S with
+eta(S).
+
+Write B_S=S/P; the measurable lift smapstoeta(s)^{-1}F_0(sP) is
+essentially left invariant, hence constant; right-P consistency makes
+it fixed by eta(P), and a minimal parabolic fixes exactly its point on
+a rank-one boundary, so the constant is tau and F_0=tau almost
+everywhere. Both sides are continuous in measure in z, so the
identity holds for every fixed label.
-Remove the external frame using a standard spectrum $Ω^+$ of the
+Remove the external frame using a standard spectrum Ω^+ of the
commutative start observation algebra, with a faithful normal
-probability class. The normal representation of $D_*=L^∞(T)$ by
-its start decomposable fields is pullback via a measurable $h(t,omega)$
-into $T$; the generator identities read $tp_z(omega)=h(t,omega)tau(z)$
-for almost all $(t,omega)$, and choosing one generic frame $t_0$ gives
-$a(omega)=t_0^{-1}h(t_0,omega)$, so $p_z(omega)=a(omega)tau(z)$.
-Each $a(omega)tau$ is continuous, extending to every fixed label; the
-end family gives $b$ the same way.
-
-On the tensor product, $a$ and $b$ commute. Substituting
+probability class. The normal representation of D_*=L^∞(T) by
+its start decomposable fields is pullback via a measurable h(t,omega)
+into T; the generator identities read tp_z(omega)=h(t,omega)tau(z)
+for almost all (t,omega), and choosing one generic frame t_0 gives
+a(omega)=t_0^{-1}h(t_0,omega), so p_z(omega)=a(omega)tau(z).
+Each a(omega)tau is continuous, extending to every fixed label; the
+end family gives b the same way.
+
+On the tensor product, a and b commute. Substituting
(eq:finite-panel-laws) in the exact graph link gives
-$atau(z)=g,btau(z)$ for countably dense labels, and since $tau$ is
-onto with faithful target action, the common zero set is $g=ab^{-1}$.
+atau(z)=g,btau(z) for countably dense labels, and since tau is
+onto with faithful target action, the common zero set is g=ab^{-1}.
Compact clopen partitions and more labels give exact graph
substitution; exhaustion and monotone approximation yield
(eq:finite-projectivity-sharp). The map from target
@@ -3749,65 +3748,65 @@ injective, with Borel inverse.
## Uniform finite-place upper speed
-For each finite-place effective panel group $R$, choose a norm on its
-prime-field adjoint Lie algebra, with $|p|=p^{-1}$ on $mathbb Q_p$,bbQ_p$, and
-let $ell_R$ be its rank-one Cartan gap. Constants $d_R>0$ depend only
+For each finite-place effective panel group R, choose a norm on its
+prime-field adjoint Lie algebra, with |p|=p^{-1} on mathbb Q_p,bbQ_p, and
+let ell_R be its rank-one Cartan gap. Constants d_R>0 depend only
on the panel type:
- log|Ad(r)|=d_Rell_R(r)+O(1),qquad rin R,
+ log|Ad(r)|=d_Rell_R(r)+O(1), rin R,
with uniform error. On a dominant split translation the largest adjoint
eigenvalue has logarithm a fixed positive multiple of the simple-root
-coordinate; converting from $|cdot|_k$ to $|cdot|_p^{[k:mathbb Q_p]}bbQ_p]$
-with the largest relative root multiple $r_{max}in{1,2}$,
-$d_R=r_{max}/[k:mathbb Q_p]$.bbQ_p]$. Bounded centralizer and compact Cartan
+coordinate; converting from |cdot|_k to |cdot|_p^{[k:mathbb Q_p]}bbQ_p]
+with the largest relative root multiple r_{max}in{1,2},
+d_R=r_{max}/[k:mathbb Q_p].bbQ_p]. Bounded centralizer and compact Cartan
factors change operator norms by bounded amounts. Symmetry under
-inversion gives $d_R$ unchanged by opposition, so the $d_R$ are drawn
+inversion gives d_R unchanged by opposition, so the d_R are drawn
from a single type list.
[Uniform finite-place upper speed]
-After the inner limit of a translated panel link, let $tge0$ be the
+After the inner limit of a translated panel link, let tge0 be the
total missing-root coordinate of the source translation, including all
slower offsets. The translation is taken on the permissible cocharacter
-mesh. If $g$ is the target residual projectivity, then
+mesh. If g is the target residual projectivity, then
d_Tell_T(g)le d_St+O_{tight}(1),
with tightness on each fixed ordinary input vector, uniformly in the
-slower translations with $t$ as center. This holds coordinatewise and
+slower translations with t as center. This holds coordinatewise and
simultaneously as an upper-tail estimate on many-panel links.
Proof.
Use the convention in which the translated start observation is
-$P_z^{(t)}=P_{s_tz}$ on its original input space and the end
-observation is $Q_z$. Thus $s_t$ is the effective start-label transport
-($t$ is on the permissible mesh, so $s_tin S$ and $eta(s_t)$ is
+P_z^{(t)}=P_{s_tz} on its original input space and the end
+observation is Q_z. Thus s_t is the effective start-label transport
+(t is on the permissible mesh, so s_tin S and eta(s_t) is
defined). Proposition~prop:transfer-residue-graph and
-Proposition~prop:finite-maps give $atau(s_tz)=g,btau(z)$, so
+Proposition~prop:finite-maps give atau(s_tz)=g,btau(z), so
g=aeta(s_t)b^{-1},
-where $s_t$ is deterministic with total missing gap $t$. Crucially
-$a,b$ are observations on the original independent input spaces, so all
+where s_t is deterministic with total missing gap t. Crucially
+a,b are observations on the original independent input spaces, so all
slower-translation dependence passes into the middle factor. Other split
translations have zero coordinate in this residue and act trivially, and
any remaining bounded effective actions range in a fixed compact set.
-With $A=deta$, Equation~(eq:finite-adjoint-intertwining) gives
-$Ad(eta(s_t))=AAd(s_t)A^{-1}$, hence
-$log|Ad(eta(s_t))|le d_St+C$, and the same for $s_t^{-1}$. Fix
-$xiinoverlineoverline II ⊗ H'$ and $ε>0$; the commuting
-$a,b$ have a joint scalar spectral measure on $xi$. Choose compact
+With A=deta, Equation~(eq:finite-adjoint-intertwining) gives
+Ad(eta(s_t))=AAd(s_t)A^{-1}, hence
+log|Ad(eta(s_t))|le d_St+C, and the same for s_t^{-1}. Fix
+xiinoverlineoverline II ⊗ H' and ε>0; the commuting
+a,b have a joint scalar spectral measure on xi. Choose compact
sets containing both observations except for squared input norm
-$leε|xi|^2$. On that joint cut $|Ad(a)|$ and
-$|Ad(b^{-1})|$ are uniformly bounded in $t$ and slower parameters;
+leε|xi|^2. On that joint cut |Ad(a)| and
+|Ad(b^{-1})| are uniformly bounded in t and slower parameters;
submultiplicativity with (eq:finite-translated-law) and
-(eq:finite-adjoint-growth) gives $d_Tell_T(g)le d_St+C_{xi,ε}$
+(eq:finite-adjoint-growth) gives d_Tell_T(g)le d_St+C_{xi,ε}
on that cut. The exact graph intertwining and link isometry bound the
excluded output by the excluded input. This proves
(eq:finite-speed) including non-simple-tensor input vectors.
@@ -3816,9 +3815,9 @@ laws and sum the excluded masses; no commutation of different input
systems is needed.
-For a finite-place non-isolated type $i$, put $s_i=d_S/d_T$. Opposition
-leaves $s_i$ unchanged, and its product around any type cycle is one
-because all $d_R$ are drawn from the same list. The isolated types left
+For a finite-place non-isolated type i, put s_i=d_S/d_T. Opposition
+leaves s_i unchanged, and its product around any type cycle is one
+because all d_R are drawn from the same list. The isolated types left
by Corollary~cor:finite-arch-separation are archimedean, treated
below.
# Length multipliers and ray lifts
@@ -3826,20 +3825,20 @@ below.
This section supplies the length estimate needed for the isolated boundary
factors: uniform ray lifts (Proposition~prop:ray-lift) combined with
-property~$(T)$ averaging give addition of rescaled spherical length laws
+property~(T) averaging give addition of rescaled spherical length laws
(Proposition~prop:ray-addition). Addition lets us later compare
repeated source segments and rule out total superlinear growth.
The isolated components are real Lie factors (Sections~sec:transfer
and~sec:finite): quaternionic hyperbolic spaces of dimension at least
-two, and the Cayley hyperbolic plane. Fix such a space $X$; normalize its
-metric so $r(g)=d(o,go)$ is the logarithm of the dilation in the
+two, and the Cayley hyperbolic plane. Fix such a space X; normalize its
+metric so r(g)=d(o,go) is the logarithm of the dilation in the
indivisible restricted root. The argument uses an entire ambient factor;
a rank-one residue in a larger factor does not supply this axis. All other
measured coordinates, including all finite places, stay in the coefficient
algebra.
-Constants may depend on $X$ and the specified smooth functions but not on
+Constants may depend on X and the specified smooth functions but not on
the number of Fourier indices or coefficient-algebra dimensions. The
analytic estimates are stated homogeneously in operator and Hilbert norms,
allowing arbitrary finite coefficient traces. Varying finite factors for
@@ -3848,141 +3847,140 @@ measure.
## Coordinates and a uniform smooth calculus
-The horospherical group is $N_b=Vtimes Z$ with
+The horospherical group is N_b=Vtimes Z with
[
- (v,z)(v',z')=(v+v',z+z'+tfrac12[v,v']),qquad
+ (v,z)(v',z')=(v+v',z+z'+tfrac12[v,v']),
delta_R(v,z)=(Rv,R^2z).
]
-Choose inner products with $< J_zv,w>=< z,[v,w]>$
-and $J_z^2=-|z|^2id$; the horizontal layer is bracket-generating. Set
-$Q=dim V+2dim Z$. In half-space coordinates $(n,lambda)in
-N_btimes(0,∞)$, with a fixed normalization of Haar measure,
+Choose inner products with < J_zv,w>=< z,[v,w]>
+and J_z^2=-|z|^2id; the horizontal layer is bracket-generating. Set
+Q=dim V+2dim Z. In half-space coordinates (n,lambda)in
+N_btimes(0,∞), with a fixed normalization of Haar measure,
ds_X^2=(dlambda^2)/(lambda^2)
+(|dv|^2)/(4lambda^2)
+(|dz-tfrac12[v,dv]|^2)/(4lambda^4),
- qquad dp=frac{dn,dlambda}{lambda^{Q+1}}.
+ dp=frac{dn,dlambda}{lambda^{Q+1}}.
These are the Iwasawa coordinates for the indicated symmetric spaces; the
-two root spaces are $V$ and $Z$. A vertical dilation has length $|log R|$.
+two root spaces are V and Z. A vertical dilation has length |log R|.
-On the compact boundary $∂ X$, the horizontal term in
+On the compact boundary ∂ X, the horizontal term in
(eq:ray-metric) determines a canonical horizontal conformal class;
we take the compact-stabilizer average of a smooth representative and let
-$d_{∂ X}$ be its Carnot--Carath'eodory distance (comparable to
-$(|v|^4/16+|z|^2)^{1/4}$ in a buffered chart).
+d_{∂ X} be its Carnot--Carath'eodory distance (comparable to
+(|v|^4/16+|z|^2)^{1/4} in a buffered chart).
-A scale box is a translate of a $R$-dilated fixed bounded box in
+A scale box is a translate of a R-dilated fixed bounded box in
exponential coordinates, with a fixed larger box available for smooth
-cutoffs. A normalized $C^m$ bound is the ordinary $C^m$ bound after
+cutoffs. A normalized C^m bound is the ordinary C^m bound after
translation and dilation to the fixed box.
[Distance and changes of scale]
-With constants uniform on buffered compact charts and in normalized $C^m$
-norms for every $mge0$:
+With constants uniform on buffered compact charts and in normalized C^m
+norms for every mge0:
[label=(roman*)]
-item If $(v,z)=log(n^{-1}n')$, then
+ If (v,z)=log(n^{-1}n'), then
cosh d((n,lambda),(n',lambda'))
=frac{√((lambda^2+lambda'^2+|v|^2/4)^2+|z|^2)}
{2lambdalambda'}.
+ Changes between the compactly rotated horospherical boundary charts
+preserve normalized bounds on boxes of every scale 0<Rle1, and in the
+adjoined height 0lelambdale CR. The conversion between the outward
+endpoint and e^{-r} (or finite half-space coordinates) has the form
-item Changes between the compactly rotated horospherical boundary charts
-preserve normalized bounds on boxes of every scale $0<Rle1$, and in the
-adjoined height $0lelambdale CR$. The conversion between the outward
-endpoint and $e^{-r}$ (or finite half-space coordinates) has the form
-
- n=n_0(e^{-2r}U,e^{-4r}W),qquad
+ n=n_0(e^{-2r}U,e^{-4r}W),
lambda=e^{-r}b,
-where $U,W,b$ are smooth in $r^{-1}$ and $e^{-r}$, with $b$ bounded above
+where U,W,b are smooth in r^{-1} and e^{-r}, with b bounded above
and below.
-item Let $g$ have length $L$, pole $g^-$ and image endpoint $g^+$. An
-input annulus of size $q=e^{-u}$ (including height) maps to an output
-annulus of size $q'=e^{-L+u}$, $0le ule L$, with normalized derivatives
-bounded. A smaller boundary box of radius $deltale cq$, height at most
-$cdelta$, has image radius at most $Ce^{2u-L}delta$. Exterior pieces
-$qge c$ have output upper bound $Ce^{-L}$.
+ Let g have length L, pole g^- and image endpoint g^+. An
+input annulus of size q=e^{-u} (including height) maps to an output
+annulus of size q'=e^{-L+u}, 0le ule L, with normalized derivatives
+bounded. A smaller boundary box of radius deltale cq, height at most
+cdelta, has image radius at most Ce^{2u-L}delta. Exterior pieces
+qge c have output upper bound Ce^{-L}.
Proof.
-(i). Translate and dilate the first point to $(0,1)$; put
-$A=lambda^2+1+|v|^2/4$, $C=A^2+|z|^2$, $c=sqrt C/(2lambda)$. In the
-orthonormal frame the three components of $nabla c$ are
-$lambda A/sqrt C-sqrt C/(2lambda)$, $(Av+J_zv)/(2sqrt C)$,
-$lambda z/sqrt C$. Since $J_zvperp v$ and $|J_zv|^2=|z|^2|v|^2$, the
-squared norms sum to $c^2-1$. Thus $arcoshc$ has gradient
+(i). Translate and dilate the first point to (0,1); put
+A=lambda^2+1+|v|^2/4, C=A^2+|z|^2, c=sqrt C/(2lambda). In the
+orthonormal frame the three components of nabla c are
+lambda A/sqrt C-sqrt C/(2lambda), (Av+J_zv)/(2sqrt C),
+lambda z/sqrt C. Since J_zvperp v and |J_zv|^2=|z|^2|v|^2, the
+squared norms sum to c^2-1. Thus arcoshc has gradient
norm one off its unique zero. It is proper, so it is the distance.
(ii). One convenient Weyl inversion in these coordinates is
j(v,z,lambda)=
(((-A_0+J_z)v)/(B_0),-frac z{B_0},
- fraclambda{√(B_0)}),quad
- A_0=lambda^2+|v|^2/4,quad B_0=A_0^2+|z|^2,
+ fraclambda{√(B_0)}),
+ A_0=lambda^2+|v|^2/4, B_0=A_0^2+|z|^2,
-with $jdelta_R=delta_{R^{-1}}j$. On a compact annulus
-$(|v|^4/16+|z|^2)^{1/4}+lambdaasymp1$, all derivatives are bounded. In
+with jdelta_R=delta_{R^{-1}}j. On a compact annulus
+(|v|^4/16+|z|^2)^{1/4}+lambdaasymp1, all derivatives are bounded. In
a finite Bruhat cell the Gauss factors of a compact rotation times a
horospherical are smooth with bounded derivatives; passing the boundary
-factor across $a_{loglambda}$ and Taylor expanding at $o$, the
-horizontal and central coordinate errors are $lambda^2U$ and
-$lambda^4W$, giving (eq:ray-collar). Boundary transitions are
+factor across a_{loglambda} and Taylor expanding at o, the
+horizontal and central coordinate errors are lambda^2U and
+lambda^4W, giving (eq:ray-collar). Boundary transitions are
H-type contact maps, so the central horizontal linear term vanishes; the
same Taylor computation applies after substitution and proves (ii).
-(iii). Compactly rotate $g$ to $delta_{e^{-L}}j$; rescale input by
-$q^{-1}$ and output by $(q')^{-1}$ to see exactly the restriction of $j$
+(iii). Compactly rotate g to delta_{e^{-L}}j; rescale input by
+q^{-1} and output by (q')^{-1} to see exactly the restriction of j
to a fixed buffered annulus. The derivative and height-ratio claims
follow from (eq:ray-inversion); the smaller-box claim is (ii) with
-relative radius $delta/q$.
+relative radius delta/q.
-The horizontal density on $∂ X$ used later is the product of the
-horizontal volume density with the density on $T∂ X/mathcal V$calV$
-induced by the surjective bracket $∧^2mathcal VcalVto
-T∂ X/mathcal V$.calV$. It is compact-invariant with mass one. Naturality
+The horizontal density on ∂ X used later is the product of the
+horizontal volume density with the density on T∂ X/mathcal VcalV
+induced by the surjective bracket ∧^2mathcal VcalVto
+T∂ X/mathcal V.calV. It is compact-invariant with mass one. Naturality
of the bracket makes it the compact angular probability, and its Jacobian
-under a horizontal dilation $dcdotIsom$ is exactly $d^Q$.
+under a horizontal dilation dcdotIsom is exactly d^Q.
[Triangle defect]
-There is $C$ such that if $x,yin X$ and
-$b=d(o,x)+d(o,y)-d(x,y)>0$, then
-$d_{∂ X}(x^+,y^+)le Ce^{-b/2}$. Consequently
-$r(g)+r(h)-r(gh)ge b>0$ forces $d_{∂ X}(g^-,h^+)le Ce^{-b/2}$.
+There is C such that if x,yin X and
+b=d(o,x)+d(o,y)-d(x,y)>0, then
+d_{∂ X}(x^+,y^+)le Ce^{-b/2}. Consequently
+r(g)+r(h)-r(gh)ge b>0 forces d_{∂ X}(g^-,h^+)le Ce^{-b/2}.
Proof.
-Both radii are at least $b/2$. In a compactly rotated finite chart
+Both radii are at least b/2. In a compactly rotated finite chart
containing both endpoints, either the boundary distance is at most the
fixed boundary-coordinate error (by (eq:ray-collar)), or
-homogeneous separation is comparable to $delta$ and (eq:ray-distance)
-gives $d(x,y)ge d(o,x)+d(o,y)+2logdelta-C$. Apply to $x=g^{-1}o$,
-$y=ho$.
+homogeneous separation is comparable to delta and (eq:ray-distance)
+gives d(x,y)ge d(o,x)+d(o,y)+2logdelta-C. Apply to x=g^{-1}o,
+y=ho.
[Smooth separation and cells]
Fix row and column dimensions and a bound on auxiliary compact variables.
-For every summable power $N$ there are $m$ and $C$ such that:
+For every summable power N there are m and C such that:
[label=(roman*)]
-item A smooth symbol on the product of two fixed buffered boxes with
-$C^m$ norm at most $A$ has a separated Fourier expansion with coefficients
-bounded by $CA(1+|nu|+|nu'|)^{-N}$; the unary factors are supported in
+ A smooth symbol on the product of two fixed buffered boxes with
+C^m norm at most A has a separated Fourier expansion with coefficients
+bounded by CA(1+|nu|+|nu'|)^{-N}; the unary factors are supported in
the larger boxes with absolute value at most one, and unary symbols with
compact external parameters work the same way.
-item If the pairs of boxes supporting a kernel form a graph of degree at
-most $b$, and normalized extensions are bounded by $A$, the kernel is a
-Schur multiplier of norm at most $C_bA$; matrix amplifications have the
+ If the pairs of boxes supporting a kernel form a graph of degree at
+most b, and normalized extensions are bounded by A, the kernel is a
+Schur multiplier of norm at most C_bA; matrix amplifications have the
same bound.
Proof.
(i) is integration by parts on a fixed torus after smooth cutoff. (ii): a
-degree-$b$ graph is a union of at most $2b-1$ matchings (greedy edge
+degree-b graph is a union of at most 2b-1 matchings (greedy edge
coloring). Each matching gives a sum of disjoint rectangular rank-one
-kernels, with $L^1$ factorization bound; overlap is split by assigning
+kernels, with L^1 factorization bound; overlap is split by assigning
each point to a bounded number of boxes. Summation and the absolutely
convergent Fourier series give (ii); matrix amplification is unaffected
by the factorization.
@@ -3991,52 +3989,52 @@ by the factorization.
## Uniform radial multipliers
[Radial cutoffs]
-Let $fin C^∞([0,∞))$ be constant near zero and near infinity.
-There are $R_f,C_f>0$ such that $(x,x')mapsto f(d(x,x')/R)$ has Schur
-norm at most $C_f$ for every $Rge R_f$. Consequently for any
-homomorphism $GammatoIsom(X)$, any normalized scalar
-cocycle $eta$, and any finite tracial von Neumann algebra $B$, the
+Let fin C^∞([0,∞)) be constant near zero and near infinity.
+There are R_f,C_f>0 such that (x,x')mapsto f(d(x,x')/R) has Schur
+norm at most C_f for every Rge R_f. Consequently for any
+homomorphism GammatoIsom(X), any normalized scalar
+cocycle eta, and any finite tracial von Neumann algebra B, the
multiplier
[
D_{f(r/R)}: B,barbar ⊗⊗ L_eta(Gamma)to
- B,barbar ⊗⊗ L_eta(Gamma),qquad
+ B,barbar ⊗⊗ L_eta(Gamma),
sum_hY_hu_h⟼sum_hf(r(h)/R)Y_hu_h
]
-has completely bounded norm at most $C_f$; the constants are independent
-of $Gamma,eta,B$ and of the homomorphism.
+has completely bounded norm at most C_f; the constants are independent
+of Gamma,eta,B and of the homomorphism.
Proof.
-Subtract $f(∞)$. Choose $0<a<b$ with $f$ equal to its value on
-$[0,2a]$ and to $0$ on $[b,∞)$. Write $s=-loglambda$,
-$s'=-loglambda'$. Since $|s-s'|le d(x,x')$, only a bounded number of
-offset bands of height blocks of length $R$ contribute; simultaneous
-dilation reduces to $|s|,|s'|le C_bR$.
-
-Take smooth radial cutoffs $chi_j(n^{-1}n')$ at radii
-$ε_j=e^{-j}$, equal to one on a smaller ball, and
-$J=lceil C'_bRrceil$. The annuli $chi_j-chi_{j+1}$ for
-$-Jle j<J$ and the innermost ball $chi_J$ cover the support of
-$f(d/R)$. Put
+Subtract f(∞). Choose 0<a<b with f equal to its value on
+[0,2a] and to 0 on [b,∞). Write s=-loglambda,
+s'=-loglambda'. Since |s-s'|le d(x,x'), only a bounded number of
+offset bands of height blocks of length R contribute; simultaneous
+dilation reduces to |s|,|s'|le C_bR.
+
+Take smooth radial cutoffs chi_j(n^{-1}n') at radii
+ε_j=e^{-j}, equal to one on a smaller ball, and
+J=lceil C'_bRrceil. The annuli chi_j-chi_{j+1} for
+-Jle j<J and the innermost ball chi_J cover the support of
+f(d/R). Put
[
H_j=log(max(ε_j,lambda,lambda')^2)/(lambdalambda').
]
-Each piece $f(d/R)-f(H_j/R)$ on its annulus, and each height increment
-$f(H_j/R)-f(H_{j-1}/R)$, has Schur norm $O(R^{-1})$ uniformly in $j,R$.
+Each piece f(d/R)-f(H_j/R) on its annulus, and each height increment
+f(H_j/R)-f(H_{j-1}/R), has Schur norm O(R^{-1}) uniformly in j,R.
This follows case-by-case on the mixed-height annuli (writing
-$(v,z)=delta_{ε_j}(V,Z)$ on the fixed separated cell, expanding
+(v,z)=delta_{ε_j}(V,Z) on the fixed separated cell, expanding
with compact auxiliary scale ratios), then at comparable heights by the
-sign-cutoff argument in $s/R-s'/R$, where $R^me^{-aR}$ is bounded on the
+sign-cutoff argument in s/R-s'/R, where R^me^{-aR} is bounded on the
smooth pieces. The telescoping identity
[
sum_{j=-J}^{J-1}phi_j(chi_j-chi_{j+1})+phi_Jchi_J
=phi_{-J}chi_{-J}+sum_{j=-J+1}^J(phi_j-phi_{j-1})chi_j
]
-with $phi_j=f(H_j/R)$ and $phi_{-J}=0$ finishes: each ball kernel has
+with phi_j=f(H_j/R) and phi_{-J}=0 finishes: each ball kernel has
uniformly bounded Schur norm by smooth separation, and there are
-$O(R)$ pieces of size $O(R^{-1})$ each.
+O(R) pieces of size O(R^{-1}) each.
-Restrict to orbit points $h^{-1}o$: the regular matrix of the Fourier
+Restrict to orbit points h^{-1}o: the regular matrix of the Fourier
multiplier is entrywise multiplication by this kernel, up to scalar
phases already in the twisted representation. The factorization proof is
unchanged, giving complete boundedness.
@@ -4044,19 +4042,19 @@ unchanged, giving complete boundedness.
## One-sided localization
-Let $D=Bbarbar ⊗⊗ L_eta(Gamma)$ with product trace $tau$ and
-Fourier coefficients $Y_hin L^2(B)$; $D_varphi$ denotes coefficient
-multiplication on $L^2(D)$. For finite Fourier sums,
-$(YY^*)_g=sum_heta(gh,h^{-1})overline{eta(h,h^{-1})}Y_{gh}Y_h^*$,
+Let D=Bbarbar ⊗⊗ L_eta(Gamma) with product trace tau and
+Fourier coefficients Y_hin L^2(B); D_varphi denotes coefficient
+multiplication on L^2(D). For finite Fourier sums,
+(YY^*)_g=sum_heta(gh,h^{-1})overline{eta(h,h^{-1})}Y_{gh}Y_h^*,
and phases have absolute value one.
[Unary Plancherel estimate]
-Let $varphi_{i,nu}:Gammatomathbb C$bbC$ satisfy
-$sup_hsum_i|varphi_{i,nu}(h)|^2le b$ for every $nu$. With $T_g$ an
-arbitrary family of indices, $i(g,t)$ chosen with at most $b$
-occurrences of each $i$ for fixed $g$, and
-$z_{g,t}=sum_nu c_{g,t,nu}[Y(D_{varphi_{i(g,t),nu}}Y)^*]_g$ where
-$|c_{g,t,nu}|le Aa_nu$, $sum_nu a_nu<∞$, we have
+Let varphi_{i,nu}:Gammatomathbb CbbC satisfy
+sup_hsum_i|varphi_{i,nu}(h)|^2le b for every nu. With T_g an
+arbitrary family of indices, i(g,t) chosen with at most b
+occurrences of each i for fixed g, and
+z_{g,t}=sum_nu c_{g,t,nu}[Y(D_{varphi_{i(g,t),nu}}Y)^*]_g where
+|c_{g,t,nu}|le Aa_nu, sum_nu a_nu<∞, we have
(sum_gsum_{tin T_g}|z_{g,t}|_{2,B}^2)^{1/2}
le bA(sum_nu a_nu)|Y|_∞|Y|_2.
@@ -4065,20 +4063,20 @@ The same holds with the first factor localized instead.
Proof.
-For one $nu$, discard admissibility after using the multiplicity bound;
+For one nu, discard admissibility after using the multiplicity bound;
Plancherel plus the operator bound for the uncut factor give
-$sum_{g,t}|[Y(D_{varphi_{i(g,t),nu}}Y)^*]_g|_2^2
-le b^2|Y|_∞^2|Y|_2^2$. Minkowski sums the absolutely summable
+sum_{g,t}|[Y(D_{varphi_{i(g,t),nu}}Y)^*]_g|_2^2
+le b^2|Y|_∞^2|Y|_2^2. Minkowski sums the absolutely summable
frequencies; adjoints give the other orientation. Only the operator norm
of the uncut factor is used.
[Angular column estimate]
-Fix $X$ and an integer $m$; there are $c,C>0$ with the following
-property. Put $delta=e^{-s}$, $sge1$. Let $(f_C)_C$ be smooth masks
-supported on buffered boundary boxes of radius $delta$, with fixed
-overlap and normalized $C^m$ bound. If $Yin D$ has Fourier support in
-${rge l}$ with $lge Cs+C$, then with $A_C=D_{f_C(h^+)}Y$,
+Fix X and an integer m; there are c,C>0 with the following
+property. Put delta=e^{-s}, sge1. Let (f_C)_C be smooth masks
+supported on buffered boundary boxes of radius delta, with fixed
+overlap and normalized C^m bound. If Yin D has Fourier support in
+{rge l} with lge Cs+C, then with A_C=D_{f_C(h^+)}Y,
sum_{C,D'}|A_CA_{D'}^*|_2^2
=Bigl|sum_CA_C^*A_CBigr|_2^2
@@ -4090,144 +4088,144 @@ scalar twists.
Proof.
-Estimate the coefficients of $A_CA_{D'}^*$ at fixed shift $g$; put
-$L=r(g)$; the weights in $YY^*$ are $f_C((gh)^+)overlineoverline{f_f_{D'}(h^+)}$.
-Only pairs with both $h$ and $gh$ in the long support matter.
+Estimate the coefficients of A_CA_{D'}^* at fixed shift g; put
+L=r(g); the weights in YY^* are f_C((gh)^+)overlineoverline{f_f_{D'}(h^+)}.
+Only pairs with both h and gh in the long support matter.
Split into nonexceptional and exceptional pairs. On a nonexceptional
-input box separated from $g^-$ by a large multiple of $delta$, with
-input pole distance $q=e^{-u}$, either $2ule L+C_1$ (then
+input box separated from g^- by a large multiple of delta, with
+input pole distance q=e^{-u}, either 2ule L+C_1 (then
Lemma~lem:ray-geometry(iii) makes the map to the output boundary
have bounded normalized derivatives, and Fourier expansion makes the
-pair weight a unary expansion in $h$ with compact-coefficient decay) or
-$2u>L+C_1$ (then inversion gives output pole distance $e^{-L+u}$ and the
-reversed unary estimate applies). Bounded $L$ and the transition
-$2u=L+O(1)$ use larger constants. Lemma~lem:ray-unary then bounds
+pair weight a unary expansion in h with compact-coefficient decay) or
+2u>L+C_1 (then inversion gives output pole distance e^{-L+u} and the
+reversed unary estimate applies). Bounded L and the transition
+2u=L+O(1) use larger constants. Lemma~lem:ray-unary then bounds
every nonexceptional pair; a map with bounded magnification sends one
-buffered $delta$-box into a ball of radius $Cdelta$, giving the
+buffered delta-box into a ball of radius Cdelta, giving the
required multiplicity.
-The exceptional pairs lie within $C_2delta$ of the two poles. On such a
+The exceptional pairs lie within C_2delta of the two poles. On such a
pair compatible with the long supports, the two current-point pole sizes
-have product comparable to $e^{-L}$; hence
+have product comparable to e^{-L}; hence
Lge 2s-C_3.
Split each mask as its value at its pole plus its difference from that
-value. The constant terms cost $C|YY^*|_2$; the mixed terms are handled
+value. The constant terms cost C|YY^*|_2; the mixed terms are handled
by Lemma~lem:ray-unary; the product of differences is split into
-smooth unit-width annuli in the pole coordinate $e^{-u}$,
-$0le ule L/2+C_4$, and on such an annulus the product seminorm obeys
+smooth unit-width annuli in the pole coordinate e^{-u},
+0le ule L/2+C_4, and on such an annulus the product seminorm obeys
C_mmin(1,e^{s-u})min(1,e^{s-L+u})le C'_m e^{-|s-u|}.
-At fixed integer depth $u$, use reference boundary boxes of radius
-$e^{-u}$ with height $e^{-u}$ as an auxiliary unary variable; only
+At fixed integer depth u, use reference boundary boxes of radius
+e^{-u} with height e^{-u} as an auxiliary unary variable; only
boundedly many such boxes meet the annulus. Lemmas~lem:ray-cells
-and~lem:ray-unary give $Ce^{-|s-u|}|Y|_∞|Y|_2$ for the
-depth-$u$ square sum; geometric summation and the reversed annuli sum
+and~lem:ray-unary give Ce^{-|s-u|}|Y|_∞|Y|_2 for the
+depth-u square sum; geometric summation and the reversed annuli sum
these uniformly. The written equality is the trace identity
-$tau(A_CA_{D'}^*A_{D'}A_C^*)=tau(A_C^*A_CA_{D'}^*A_{D'})$.
+tau(A_CA_{D'}^*A_{D'}A_C^*)=tau(A_C^*A_CA_{D'}^*A_{D'}).
-For general $Y$, take bounded Fourier-polynomial approximations with a
-smooth radial cutoff equal to one on ${rge l}$ and zero on
-${rle l/2}$; Proposition~prop:ray-radial bounds the approximants
-uniformly. Increase the constant in $l$ by the slack. Each masked column
-converges in $L^2$ and each product of two in $L^1$ by tracial
+For general Y, take bounded Fourier-polynomial approximations with a
+smooth radial cutoff equal to one on {rge l} and zero on
+{rle l/2}; Proposition~prop:ray-radial bounds the approximants
+uniformly. Increase the constant in l by the slack. Each masked column
+converges in L^2 and each product of two in L^1 by tracial
Cauchy--Schwarz; for a fixed finite family the diagonal-terms bound in
-the Hilbert direct sum supplies weak $L^2$ compactness, and the $L^1$
+the Hilbert direct sum supplies weak L^2 compactness, and the L^1
convergence identifies the limit. This makes every masked column live in
-$L^4$, giving the stated trace identity for finite families; exhaustion
+L^4, giving the stated trace identity for finite families; exhaustion
finishes.
## The ray lift and its densities
-Let $mathcal K_X=L^2(X,dp)$calK_X=L^2(X,dp)$ and let $T_g$ be translation on
-$mathcal K_X$.calK_X$. Define
+Let mathcal K_X=L^2(X,dp)calK_X=L^2(X,dp) and let T_g be translation on
+mathcal K_X.calK_X. Define
[
- Delta:D→ B(mathcal K_X)calK_X)barbar ⊗⊗ D,qquad
+ Delta:D→ B(mathcal K_X)calK_X)barbar ⊗⊗ D,
Delta(bu_h)=T_h ⊗ bu_h,
]
a normal unital homomorphism (conjugate by the unitary sending
-$zeta ⊗ delta_h$ to $T_h^{-1}zeta ⊗ delta_h$).
+zeta ⊗ delta_h to T_h^{-1}zeta ⊗ delta_h).
-Choose real smooth $psi,F$ with $supppsiSubset(1,2)$, $F$
-compactly supported on $N_b$ and invariant under the compact isotropy
-of a ray, with unit $L^2$ norms. For $d_1ge1$, basepoint $x$, and
-ideal endpoint $v$, use a chart sending $v$ to infinity and put
+Choose real smooth psi,F with supppsiSubset(1,2), F
+compactly supported on N_b and invariant under the compact isotropy
+of a ray, with unit L^2 norms. For d_1ge1, basepoint x, and
+ideal endpoint v, use a chart sending v to infinity and put
theta_{x,d_1}(p,v)=d_1^{-1/2}
psi((log(lambda_p/lambda_x))/(d_1))
F(delta_{lambda_p^{-1}}(n_x^{-1}n_p)).
-This is equivariant in $x,p,v$ and has unit $L^2$ norm, supported in a
-bounded neighborhood of the ray from $x$ to $v$ at depths
-$[d_1-O(1),2d_1+O(1)]$. The coefficient isometry is
-$mathbb V_bbV_{d_1}Y=sum_htheta_{o,d_1}(cdot,h^+) ⊗ Y_hu_h$.
+This is equivariant in x,p,v and has unit L^2 norm, supported in a
+bounded neighborhood of the ray from x to v at depths
+[d_1-O(1),2d_1+O(1)]. The coefficient isometry is
+mathbb V_bbV_{d_1}Y=sum_htheta_{o,d_1}(cdot,h^+) ⊗ Y_hu_h.
-For a column $xiinmathcal K_XcalK_X ⊗ L^2(D)$, the left and right
-densities $rho^L_xi,rho^R_xiin L^1(D)$ represent the vector
-functionals for $Delta(D)$ and $D^{op}$:
-$<xi,Delta(x)xi>=tau(rho^L_xi x)$,
-$<xi,xi x>=tau(rho^R_xi x)=xi^*xi$.
+For a column xiinmathcal K_XcalK_X ⊗ L^2(D), the left and right
+densities rho^L_xi,rho^R_xiin L^1(D) represent the vector
+functionals for Delta(D) and D^{op}:
+<xi,Delta(x)xi>=tau(rho^L_xi x),
+<xi,xi x>=tau(rho^R_xi x)=xi^*xi.
[Ray lift]
Take any sequence of finite tracial coefficient algebras, twists, and
-countable homomorphisms to $Isom(X)$, with
-$d_1to∞$, $l/d_1to∞$, and $Yin D$ Fourier-supported in
-${rge l}$.
+countable homomorphisms to Isom(X), with
+d_1to∞, l/d_1to∞, and Yin D Fourier-supported in
+{rge l}.
[label=(roman*)]
-item For large indices, $mathbb V_bbV_{d_1}Y$ has left density for
-$Delta$ on $D$ and right density on $D^{op}$, both in $L^2$
-with norm at most $C|Y|_∞|Y|_2$; also
-$|mathbb V_bbV_{d_1}Y(mathbb V_bbV_{d_1}Y)^*|_{2,Tr ⊗ tau}$ has this
+ For large indices, mathbb V_bbV_{d_1}Y has left density for
+Delta on D and right density on D^{op}, both in L^2
+with norm at most C|Y|_∞|Y|_2; also
+|mathbb V_bbV_{d_1}Y(mathbb V_bbV_{d_1}Y)^*|_{2,Tr ⊗ tau} has this
bound.
-item If $suppwidehatwidehat xxsubset{rle a}$ and $a/d_1to0$, then
-$|mathbb V_bbV_{d_1}(xY)-Delta(x)mathbb V_bbV_{d_1}Y|_2to0$ and
-$|mathbb V_bbV_{d_1}(Yx)-(mathbb V_bbV_{d_1}Y)x|_2to0$.
-item For another long column $Z$ and bounded $x$, if for every
-contributing triple $hinsuppwidehatwidehat Y$,Y$, $hkinsuppwidehatwidehat Z$,Z$,
-$kinsuppwidehatwidehat x$x$ one has
-$d_{∂ X}(h^+,(hk)^+)le e^{-cl'}$ with $c>0$,
-$l'/d_1to∞$, then
+ If suppwidehatwidehat xxsubset{rle a} and a/d_1to0, then
+|mathbb V_bbV_{d_1}(xY)-Delta(x)mathbb V_bbV_{d_1}Y|_2to0 and
+|mathbb V_bbV_{d_1}(Yx)-(mathbb V_bbV_{d_1}Y)x|_2to0.
+ For another long column Z and bounded x, if for every
+contributing triple hinsuppwidehatwidehat Y,Y, hkinsuppwidehatwidehat Z,Z,
+kinsuppwidehatwidehat xx one has
+d_{∂ X}(h^+,(hk)^+)le e^{-cl'} with c>0,
+l'/d_1to∞, then
<mathbb V_bbV_{d_1}Z,(mathbb V_bbV_{d_1}Y)x>
-< Z,Yx>→0,
-uniformly with $|x|_∞,|Y|_2,|Z|_2$ bounded.
+uniformly with |x|_∞,|Y|_2,|Z|_2 bounded.
All statements hold for matrix versions and measurable fields.
Proof. [Right density]
-Write $K_{d_1}(v,w)=inttheta_{o,d_1}(p,v)theta_{o,d_1}(p,w),dp$
-and disintegrate at radial depth $s=d(o,p)in[d_1-C,2d_1+C]$. With
-measure $ds/d_1$, the kernels $k_{d_1,s}$ sit on adjacent boundary
-boxes of radius $e^{-s}$ with normalized $C^m$ bounds independent of
-$d_1,s$. The substitution $n_p=n_xdelta_{lambda_p}q$,
-$t=log(lambda_p/lambda_x)/d_1$ fixes the normalization; polar volume
-$J(s)asymp e^{Qs}$ compensates the angular shadow $O(e^{-Qs})$, and
+Write K_{d_1}(v,w)=inttheta_{o,d_1}(p,v)theta_{o,d_1}(p,w),dp
+and disintegrate at radial depth s=d(o,p)in[d_1-C,2d_1+C]. With
+measure ds/d_1, the kernels k_{d_1,s} sit on adjacent boundary
+boxes of radius e^{-s} with normalized C^m bounds independent of
+d_1,s. The substitution n_p=n_xdelta_{lambda_p}q,
+t=log(lambda_p/lambda_x)/d_1 fixes the normalization; polar volume
+J(s)asymp e^{Qs} compensates the angular shadow O(e^{-Qs}), and
normalized derivatives are bounded by translating back along rays and
-rescaling neighbors by $e^{-s}$.
+rescaling neighbors by e^{-s}.
-Separate $k_{d_1,s}$ on adjacent boxes: at each frequency the right
-density is $sum t_{C,D'}A_C^*B_{D'}$ with scalar matrix $t$ of bounded
+Separate k_{d_1,s} on adjacent boxes: at each frequency the right
+density is sum t_{C,D'}A_C^*B_{D'} with scalar matrix t of bounded
degree and operator norm, and both mask families meeting the
hypotheses of Lemma~lem:ray-angular. Tracial H"older plus
-(eq:input-7) give $le C|Y|_∞|Y|_2$; Fourier
+(eq:input-7) give le C|Y|_∞|Y|_2; Fourier
coefficients are summable, chosen above the polynomial derivative
growth of masks. The matrix assertion follows from
-$|mathbb V_bbV_{d_1}Y(mathbb V_bbV_{d_1}Y)^*|_2^2
-=|(mathbb V_bbV_{d_1}Y)^*mathbb V_bbV_{d_1}Y|_2^2$ and monotone
+|mathbb V_bbV_{d_1}Y(mathbb V_bbV_{d_1}Y)^*|_2^2
+=|(mathbb V_bbV_{d_1}Y)^*mathbb V_bbV_{d_1}Y|_2^2 and monotone
approximation.
[Common-endpoint overlap]
-For a displacement $g$ of length $L$, put
-$w_{d_1}(g,v)=int_Xtheta_{o,d_1}(p,v)theta_{go,d_1}(p,v),dp$.
-There are smooth annular partitions about $g^+$ at integer depths
-$0le jle L$ (with exterior piece and innermost ball) on which
+For a displacement g of length L, put
+w_{d_1}(g,v)=int_Xtheta_{o,d_1}(p,v)theta_{go,d_1}(p,v),dp.
+There are smooth annular partitions about g^+ at integer depths
+0le jle L (with exterior piece and innermost ball) on which
|w_{d_1}(g,cdot)-a_{d_1}(j,L)|_{C^m_{rm normalized}}
@@ -4236,66 +4234,66 @@ $0le jle L$ (with exterior piece and innermost ball) on which
psi((s-2j+L)/d_1),ds,
-with constants uniform in $L,d_1ge1$, and
+with constants uniform in L,d_1ge1, and
- |∂_ja_{d_1}(j,L)|le C/d_1,qquad
+ |∂_ja_{d_1}(j,L)|le C/d_1,
a_{d_1}(0,L)=1+O(min(1,L/d_1)).
Proof.
-Put $go=(0,ε)$, $ε=e^{-L}$, and $g^+=0$. For finite
-$v=(V,W)$, use
-$p(v)=|V|^2/4$, $D_h(v)=(h^2+p(v))^2+|W|^2$, and
-$P_h(v)=((h^2+p(v)+J_W)V/D_h,W/D_h)$. Then
-$delta_{√(D_1)}L_{P_1^{-1}}jL_{v^{-1}}$ fixes $o$ and sends $v$
-to infinity, and the displaced point is $(n,e^beta)$ with
-
- n=delta_{√(D_1)}(P_1^{-1}P_ε),qquad
+Put go=(0,ε), ε=e^{-L}, and g^+=0. For finite
+v=(V,W), use
+p(v)=|V|^2/4, D_h(v)=(h^2+p(v))^2+|W|^2, and
+P_h(v)=((h^2+p(v)+J_W)V/D_h,W/D_h). Then
+delta_{√(D_1)}L_{P_1^{-1}}jL_{v^{-1}} fixes o and sends v
+to infinity, and the displaced point is (n,e^beta) with
+
+ n=delta_{√(D_1)}(P_1^{-1}P_ε),
e^beta=ε√(D_1/D_ε).
-Put $v=delta_Ru$, $R=e^{-j}$, $t=ε/R$, and define
-$Delta_R=(1+R^2p(u))^2+R^4|W|^2$,
-$Q_R=(R^2(1+R^2p+R^2J_W)V/Delta_R,R^4W/Delta_R)$. Then
+Put v=delta_Ru, R=e^{-j}, t=ε/R, and define
+Delta_R=(1+R^2p(u))^2+R^4|W|^2,
+Q_R=(R^2(1+R^2p+R^2J_W)V/Delta_R,R^4W/Delta_R). Then
- beta=2j-L+B(R,t,u),quad
- B=tfrac12log(Delta_R/D_t),quad
+ beta=2j-L+B(R,t,u),
+ B=tfrac12log(Delta_R/D_t),
delta_Rn=m(R,t,u)=delta_{√(Delta_R)}(Q_R^{-1}P_t).
-All derivatives of $B,m$ are bounded on the buffered annulus, and
-$mto P_0(u)ne e$ as $R,tto0$. The rest follows by writing
-$H(m)=int F(q)F(m^{-1}q),dq$ (smooth, compact, $H(e)=1$):
+All derivatives of B,m are bounded on the buffered annulus, and
+mto P_0(u)ne e as R,tto0. The rest follows by writing
+H(m)=int F(q)F(m^{-1}q),dq (smooth, compact, H(e)=1):
w_{d_1}(g,v)=intpsi(t)psi(t-beta/d_1)H(delta_{e^{-td_1}}n),dt.
-Replace $B$ by $0$ using the mean-value formula (extra factor $d_1^{-1}$
-on each annular piece, using the support of $H$ and separation of $m$
-from $e$); replace $H(delta_{e^{j-s}}m)$ by
-$1_{s>j}$ using Taylor's formula for $s>j$ and the support of
-$H$ for $sle j$. The $O(1/d_1)$ derivative bounds follow similarly.
+Replace B by 0 using the mean-value formula (extra factor d_1^{-1}
+on each annular piece, using the support of H and separation of m
+from e); replace H(delta_{e^{j-s}}m) by
+1_{s>j} using Taylor's formula for s>j and the support of
+H for sle j. The O(1/d_1) derivative bounds follow similarly.
Proof. [Left density]
-At shift $g$, the left density has the coefficients of $YY^*$
-weighted by the overlap of the two rays based at $o,go$ toward
-$(gh)^+,gh^+$; only $r(g)le4d_1+C$ contributes. Replace the second
-endpoint by $(gh)^+$ with coefficient error at most
+At shift g, the left density has the coefficients of YY^*
+weighted by the overlap of the two rays based at o,go toward
+(gh)^+,gh^+; only r(g)le4d_1+C contributes. Replace the second
+endpoint by (gh)^+ with coefficient error at most
Ce^{C'd_1-l}|Y|_∞|Y|_2,
by a compact-polar computation with auxiliary variable
-$e^{l-r(h)}$ plus fixed-derivative bounds of isometries of length
-$le4d_1+C$ (product-form derivative growth at most
-$e^{C_m d_1}$) and the same volume bounds.
+e^{l-r(h)} plus fixed-derivative bounds of isometries of length
+le4d_1+C (product-form derivative growth at most
+e^{C_m d_1}) and the same volume bounds.
At each annular depth the overlap error is then a unary symbol of
-seminorm $O(d_1^{-1})$ on boundedly many reference boxes, giving
-$O(d_1^{-1})|Y|_∞|Y|_2$ for the coefficient $L^2$ norm by
-Lemma~lem:ray-unary; telescope $a_{d_1}(j,L)$ against single
+seminorm O(d_1^{-1}) on boundedly many reference boxes, giving
+O(d_1^{-1})|Y|_∞|Y|_2 for the coefficient L^2 norm by
+Lemma~lem:ray-unary; telescope a_{d_1}(j,L) against single
balls using (eq:ray-overlap-increments), and observe the at
-most $C(1+d_1)$ levels. Summing plus (eq:ray-endpoint-error)
+most C(1+d_1) levels. Summing plus (eq:ray-endpoint-error)
gives the bound. The stronger short-coefficient version
-$a=o(d_1)$ gives difference bounded in $L^2$ by
+a=o(d_1) gives difference bounded in L^2 by
C((1+a)/d_1+e^{C'd_1-l})|Y|_∞|Y|_2.
@@ -4304,113 +4302,113 @@ Polarization extends to
<mathbb V_bbV_{d_1}Z,Delta(x)mathbb V_bbV_{d_1}Y>
-< Z,xY>→0
-uniformly when norms of $Z,Y,x$ are bounded with
-$suppwidehatwidehat xxsubset{rle a}$. Bounded approximants and
-weakly convergent $L^2$ subsequences with slacked cutoffs handle
+uniformly when norms of Z,Y,x are bounded with
+suppwidehatwidehat xxsubset{rle a}. Bounded approximants and
+weakly convergent L^2 subsequences with slacked cutoffs handle
infinite Fourier support.
Proof. [Pairing (iii) and intertwining (ii)]
-$K_{d_1}(v,v)=1$; with $d(o,p)le2d_1+C$ and fixed-derivative
+K_{d_1}(v,v)=1; with d(o,p)le2d_1+C and fixed-derivative
uniform bounds, Taylor at the diagonal on adjacent boxes of radius
-$e^{-d_2}$ gives
+e^{-d_2} gives
|K_{d_1}-1|_{C^m_{rm normalized}}le C_m e^{C_m d_1-d_2},
-positive-derivative terms contributing $e^{-d_2}$, central ones
-$e^{-2d_2}$. Choose $d_1ll d_2ll l'$. Separate $K_{d_1}-1$ at
-scale $e^{-d_2}$; Cauchy--Schwarz for the cut columns bounds the
-fixed-frequency pairing by $|t||x|_∞
-(sum|Z_C|_2^2)^{1/2}(sum|Y_{D'}|_2^2)^{1/2}$, and the
+positive-derivative terms contributing e^{-d_2}, central ones
+e^{-2d_2}. Choose d_1ll d_2ll l'. Separate K_{d_1}-1 at
+scale e^{-d_2}; Cauchy--Schwarz for the cut columns bounds the
+fixed-frequency pairing by |t||x|_∞
+(sum|Z_C|_2^2)^{1/2}(sum|Y_{D'}|_2^2)^{1/2}, and the
matrix norm times the frequency sum vanishes. This proves
(eq:ray-close-pairing). For short right multiplication,
-$r(h),r(hk)ge l-a$ and $r(k)le a$ give
-$d_{∂ X}(h^+,(hk)^+)le Ce^{-c(l-2a)}$ with fixed $c>0$, so
+r(h),r(hk)ge l-a and r(k)le a give
+d_{∂ X}(h^+,(hk)^+)le Ce^{-c(l-2a)} with fixed c>0, so
(iii) applies. Expanding
-$|mathbb V_bbV_{d_1}(Yx)-(mathbb V_bbV_{d_1}Y)x|_2^2$ finishes (ii) for
-the right; for the left use (eq:ray-left-pairing) with $x^*x$
-and $Z=xY$.
+|mathbb V_bbV_{d_1}(Yx)-(mathbb V_bbV_{d_1}Y)x|_2^2 finishes (ii) for
+the right; for the left use (eq:ray-left-pairing) with x^*x
+and Z=xY.
[Stability under Hilbert-small changes]
-If $Y$ is long with bounded $|Y|_∞,|Y|_2$ and
-$x-x'to0$ in $L^2(D)$ with $x,x'$ uniformly bounded, then
-$|Delta(x-x')mathbb V_bbV_{d_1}Y|_2to0$ and
-$|(mathbb V_bbV_{d_1}Y)(x-x')|_2to0$; the analogous amplified
-statement holds in the semifinite $L^2$ norm.
+If Y is long with bounded |Y|_∞,|Y|_2 and
+x-x'to0 in L^2(D) with x,x' uniformly bounded, then
+|Delta(x-x')mathbb V_bbV_{d_1}Y|_2to0 and
+|(mathbb V_bbV_{d_1}Y)(x-x')|_2to0; the analogous amplified
+statement holds in the semifinite L^2 norm.
Proof.
-Pair the corresponding density with $(x-x')^*(x-x')$ or its
-reverse; tracial Cauchy--Schwarz with $z=x-x'$ gives
+Pair the corresponding density with (x-x')^*(x-x') or its
+reverse; tracial Cauchy--Schwarz with z=x-x' gives
max{|Delta(z)mathbb V_bbV_{d_1}Y|_2^2,
|(mathbb V_bbV_{d_1}Y)z|_2^2}
le C|Y|_∞|Y|_2|z|_∞|z|_2.
-The amplified version uses $mathbb V_bbV_{d_1}Y(mathbb V_bbV_{d_1}Y)^*$
+The amplified version uses mathbb V_bbV_{d_1}Y(mathbb V_bbV_{d_1}Y)^*
and the semifinite norm. For fields, integrate the pointwise bound
and apply Cauchy--Schwarz on the base.
## Admissible representations and spherical laws
-Fix an isolated measured factor $T$ of $G$, its symmetric space
-$X_T$, basepoint $o_T$, and $r_T(h)=d(o_T,h_T o_T)$ (normalized at
-the indivisible simple root). $D_f$ denotes the Fourier multiplier
-with symbol $f$. Finite factor traces are normalized.
+Fix an isolated measured factor T of G, its symmetric space
+X_T, basepoint o_T, and r_T(h)=d(o_T,h_T o_T) (normalized at
+the indivisible simple root). D_f denotes the Fourier multiplier
+with symbol f. Finite factor traces are normalized.
-An admissible representation for $T$ is a finite factor
-$D=Bbarbar ⊗⊗ L_eta(Gamma)$ together with a normalized
-projective representation $Θ:Gammatomathcal U(D)$calU(D)$ such that
-$DcapΘ(Gamma)'=mathbb C1$bbC1$ and the representation
+An admissible representation for T is a finite factor
+D=Bbarbar ⊗⊗ L_eta(Gamma) together with a normalized
+projective representation Θ:Gammatomathcal U(D)calU(D) such that
+DcapΘ(Gamma)'=mathbb C1bbC1 and the representation
gamma⟼AdDelta_T(Θ(gamma))
- quadhbox{on}quad
+ quadhbox{on}
L^2(B(L^2(X_T))barbar ⊗⊗ D,Tr ⊗ tau)
has no invariant vector. Here
-$Delta_T(u_h)=lambda_{X_T}(h_T) ⊗ u_h$. The scalar multiplier
-of $Θ$ need not equal $eta$.
+Delta_T(u_h)=lambda_{X_T}(h_T) ⊗ u_h. The scalar multiplier
+of Θ need not equal eta.
-For $ain G$ put $X^Θ_a(y)=Θ(c(a,y))$ (the section and
+For ain G put X^Θ_a(y)=Θ(c(a,y)) (the section and
cocycle of Section~sec:setup). For an isolated source factor
-$S$ and positive split translation $a_t$ of length $t$, the spherical
-law at $t$, on scale $R>0$, is the probability
+S and positive split translation a_t of length t, the spherical
+law at t, on scale R>0, is the probability
P_{Θ,R}(t)(E)=int_Y
bigl|D_{1_{{r_T/Rin E}}}X^Θ_{a_t}(y)bigr|_2^2,dy,
-for Borel $Esubset[0,∞]$.
+for Borel Esubset[0,∞].
[Uniform short approximation]
-Let $R_nto∞$ and $Θ_n$ be projective representations in
+Let R_nto∞ and Θ_n be projective representations in
finite tensors, with
|D_{1_{{r_T>ε R_n}}}Θ_n(gamma)|_2to0
- quad(gammainGamma, ε>0).
+ (gammainGamma, ε>0).
-Then there are $a_n=o(R_n)$ and uniformly operator-bounded fields
-$Z_{n,s}$ with Fourier support in ${r_Tle a_n}$ such that
+Then there are a_n=o(R_n) and uniformly operator-bounded fields
+Z_{n,s} with Fourier support in {r_Tle a_n} such that
sup_{sin C}|X^{Θ_n}_s-Z_{n,s}|_{L^2(Y;L^2(D_n))}to0
-for every compact $Csubset G$. Thus compact pre- and post-translations
+for every compact Csubset G. Thus compact pre- and post-translations
do not affect any limiting rescaled length law.
Proof.
-For fixed compact $C$, outside a set of measure at most
-$2ε$ both $d_0(y),d_0(sy)in K$, so
-$c(s,y)inGammacap K^{-1}CK$, a finite set; apply
+For fixed compact C, outside a set of measure at most
+2ε both d_0(y),d_0(sy)in K, so
+c(s,y)inGammacap K^{-1}CK, a finite set; apply
(eq:ray-short-fixed) to it. Smooth cutoffs
-$b_n=o(R_n)$ with $a_n=2b_n$ and Proposition~prop:ray-radial
+b_n=o(R_n) with a_n=2b_n and Proposition~prop:ray-radial
give the bounds. The radial multiplier uses
- |r_T(gh)-r_T(h)|le r_T(g),qquad |r_T(hg)-r_T(h)|le r_T(g)
+ |r_T(gh)-r_T(h)|le r_T(g), |r_T(hg)-r_T(h)|le r_T(g)
to show a short multiplier cannot connect two closed rescaled radial
intervals at positive distance; smooth radial cuts then have
@@ -4420,30 +4418,30 @@ uniformly operator-bounded.
[Averaging with a uniform gap]
There is a compactly supported, absolutely continuous, symmetric
-probability $m_0$ on $G$ and $0<c_0<1$ such that its convolution
+probability m_0 on G and 0<c_0<1 such that its convolution
operator is positive in every unitary representation and has norm at
-most $c_0$ on the orthogonal complement of the invariant vectors.
+most c_0 on the orthogonal complement of the invariant vectors.
Proof.
-Take a compact Kazhdan pair $(C,ε)$ for the product
-$G$; a symmetric compactly supported density $nu$ that is bounded
-below on $Vcup CV$ and their inverses, for a compact neighborhood
-$V$ of $1$, satisfies
-$sup_{gin C}|pi(g)xi-xi|^2le
-C_1int|pi(h)xi-xi|^2dnu(h)$, giving
-$|pi(nu)|le1-delta$ on the complement of invariants. Then
-$(1+pi(nu))/2$ is positive with the same bound, and convolving on
-both sides by a symmetric compactly supported probability $chi$
+Take a compact Kazhdan pair (C,ε) for the product
+G; a symmetric compactly supported density nu that is bounded
+below on Vcup CV and their inverses, for a compact neighborhood
+V of 1, satisfies
+sup_{gin C}|pi(g)xi-xi|^2le
+C_1int|pi(h)xi-xi|^2dnu(h), giving
+|pi(nu)|le1-delta on the complement of invariants. Then
+(1+pi(nu))/2 is positive with the same bound, and convolving on
+both sides by a symmetric compactly supported probability chi
gives a positive operator
-$pi(chi)(1+pi(nu))pi(chi)/2$ with the same bound.
+pi(chi)(1+pi(nu))pi(chi)/2 with the same bound.
[Addition in distribution]
-Let $R_nto∞$, $Θ_n$ admissible for $T$ satisfying
+Let R_nto∞, Θ_n admissible for T satisfying
(eq:ray-short-fixed). Fix an isolated source factor and a
-positive split axis. For any $t_n,u_nge0$, if $P_{Θ_n,R_n}(t_n)$
-converges weakly to $P_t$ and $P_{Θ_n,R_n}(u_n)$ to $P_u$, then
+positive split axis. For any t_n,u_nge0, if P_{Θ_n,R_n}(t_n)
+converges weakly to P_t and P_{Θ_n,R_n}(u_n) to P_u, then
P_{Θ_n,R_n}(t_n+u_n)→ P_t*P_u.
@@ -4452,82 +4450,82 @@ The same holds with compact pre- and post-translations.
Proof. [Sketch]
Pass to a subsequence with the sum laws converging. Insert a fixed
-translation $s$ between the two positive axial segments; away from
+translation s between the two positive axial segments; away from
the Bruhat wall (a proper lower-dimensional cell of measure zero),
-Cartan multiplication expresses $a_{t_n}sa_{u_n}=a_{t_n+u_n}$ up to
+Cartan multiplication expresses a_{t_n}sa_{u_n}=a_{t_n+u_n} up to
compact pre/post translations; Lemma~lem:ray-compactshort
identifies the uncut law of the inserted product with the limiting
sum.
Replacing the actual transporter by its short approximation, average
-$s$ over $m_0^{*k}$. For bounded smooth radial cuts
-$A_{n,f}=D_{f(r_T/R_n)}X^{Θ_n}_{a_{t_n}}$,
-$B_{n,g}=D_{g(r_T/R_n)}X^{Θ_n}_{a_{u_n}}(a_{u_n}^{-1}cdot)$,
-the mixed Gram forms are pairings between $A_{n,f'}^*A_{n,f}$ and
-the translate of $B_{n,g}B_{n,g'}^*$ under the induced trace action.
-Its invariant space consists of scalars; averaging with $m_0^{*k}$
-gives a uniform $O(c_0^k)$ gap. Two successive Hilbert limits give an
+s over m_0^{*k}. For bounded smooth radial cuts
+A_{n,f}=D_{f(r_T/R_n)}X^{Θ_n}_{a_{t_n}},
+B_{n,g}=D_{g(r_T/R_n)}X^{Θ_n}_{a_{u_n}}(a_{u_n}^{-1}cdot),
+the mixed Gram forms are pairings between A_{n,f'}^*A_{n,f} and
+the translate of B_{n,g}B_{n,g'}^* under the induced trace action.
+Its invariant space consists of scalars; averaging with m_0^{*k}
+gives a uniform O(c_0^k) gap. Two successive Hilbert limits give an
isometry
- J:L^2(P_t) ⊗ L^2(P_u)→mathcal K,calK,qquad
+ J:L^2(P_t) ⊗ L^2(P_u)→mathcal K,calK,
f ⊗ g⟼(A_{n,f}(sy)X^{Θ_n}_s(y)B_{n,g}(y))^{lim}.
For a positive finite output window strictly above the sum of the
inputs, (eq:ray-triangle) plus short approximation excludes it.
For the strict-deficit case, in the shortening test
-$A(sy)^*$ pairs against $X^{Θ_n}_s(y)B(y)C(y,s)$; convert to
+A(sy)^* pairs against X^{Θ_n}_s(y)B(y)C(y,s); convert to
tau(zeta_n(sy)^*Delta_T(X^{Θ_n}_s(y))xi_n(y)C(y,s))
-with $xi_n=mathbb V_bbV_{d_n}B$, $zeta_n=mathbb V_bbV_{d_n}(A^*)$. The
+with xi_n=mathbb V_bbV_{d_n}B, zeta_n=mathbb V_bbV_{d_n}(A^*). The
exact trace inequality (via Cauchy--Schwarz on the two density fields,
each with uniformly bounded trace-Hilbert norm by
-Proposition~prop:ray-lift) is bounded by $C c_0^k$ because the
+Proposition~prop:ray-lift) is bounded by C c_0^k because the
combined amplified action has no invariant vector (admissibility).
-For a short change $z$, the error is at most
-$|z|_∞|z|_2|rho^L_{xi_n}|_2$, i.e.
+For a short change z, the error is at most
+|z|_∞|z|_2|rho^L_{xi_n}|_2, i.e.
|Delta_T(z)xi_n|^2
le|z|_∞|z|_2|rho^L_{xi_n}|_2.
-Limit $nto∞$ then $kto∞$.
+Limit nto∞ then kto∞.
-For input values $0$ and $+∞$, use triangle inequalities and
+For input values 0 and +∞, use triangle inequalities and
short approximations with smooth tail cutoffs (Proposition~prop:ray-radial);
the lifted shortening argument extends because it uses positive
lower length bounds and a uniform deficit but no upper bound.
The output-measurement identification uses the exact spectral
-argument: with $a(s,t)=s+t$ on the compact square, product tests
-supported on rectangles whose closures lie in $a^{-1}(F)^c$ have
-dense span in $ran P_{rm in}(a^{-1}(F)^c)$, so
-$E(F)J P_{rm in}(a^{-1}(F)^c)=0$ and
-$E(F)J=E(F)J P_{rm in}(a^{-1}(F))$. A closed-set exhaustion
-$K_j↑ F^c$ gives $E(K_j)JP_{rm in}(a^{-1}(F))=0$, so
-$E(F^c)JP_{rm in}(a^{-1}(F))=0$; together,
-$E(F)J=J P_{rm in}(a^{-1}(F))$. Monotone classes and spectral
+argument: with a(s,t)=s+t on the compact square, product tests
+supported on rectangles whose closures lie in a^{-1}(F)^c have
+dense span in ran P_{rm in}(a^{-1}(F)^c), so
+E(F)J P_{rm in}(a^{-1}(F)^c)=0 and
+E(F)J=E(F)J P_{rm in}(a^{-1}(F)). A closed-set exhaustion
+K_j↑ F^c gives E(K_j)JP_{rm in}(a^{-1}(F))=0, so
+E(F^c)JP_{rm in}(a^{-1}(F))=0; together,
+E(F)J=J P_{rm in}(a^{-1}(F)). Monotone classes and spectral
integration give
-$E(varphi)J=JM_{varphi(s+t)}$ for $varphiin C([0,∞])$.
+E(varphi)J=JM_{varphi(s+t)} for varphiin C([0,∞]).
Convergence on the whole sequence follows by compactness of
-probability measures on $[0,∞]$.
+probability measures on [0,∞].
# Linear subsequences and heights for isolated components
At an isolated type, Section~sec:transfer has already supplied
-sharp boundary measurements $xmapsto P_x$. We now construct a height
+sharp boundary measurements xmapsto P_x. We now construct a height
at each source label: a positive injective self-adjoint operator
-commuting with $P_x$ and transforming by the inverse boundary
+commuting with P_x and transforming by the inverse boundary
Jacobians. The construction takes an inverse power of the modulus of a
-horizontal derivative of $xmapsto P_x$; the main analytic requirement
+horizontal derivative of xmapsto P_x; the main analytic requirement
is a square-integrable derivative on a dense invariant domain.
The first part controls Fourier lengths: along long source intervals,
every fixed input retains positive mass below a linear length bound.
The addition theorem (Proposition~prop:ray-addition) plus
-property~$(T)$ rules out total superlinear growth. Lattice packing then
+property~(T) rules out total superlinear growth. Lattice packing then
costs one factor of the length, canceled by averaging over a source
stabilizer interval of comparable length. The resulting estimate for
ordered products at opposite source labels controls small horizontal
@@ -4535,7 +4533,7 @@ differences; we then build the closed derivative modulus and prove its
kernel is zero.
The source ray lies in an entire simple factor. Isolated factors are
-real and locally $Sp(n,1)$, $nge2$, or $F_{4(-20)}$. Complementary
+real and locally Sp(n,1), nge2, or F_{4(-20)}. Complementary
factors may be archimedean or nonarchimedean; a compact window in those
coordinates is retained in each lattice count. Horizontal metrics and
angular probabilities are the compatible compact-invariant choices of
@@ -4543,7 +4541,7 @@ Section~sec:ray.
We use the admissibility and statistical conventions of
Definition~def:ray-admissible. All varying finite factor traces
-are normalized, $Y=G/Gamma$ has probability measure, and extra finite
+are normalized, Y=G/Gamma has probability measure, and extra finite
matrices carry the unnormalized matrix trace with fixed size. Each
application of the ray estimates has both the stated operator bounds
and bounded integrated Hilbert norms; a lower effect bound concerns a
@@ -4567,20 +4565,20 @@ not new assumptions.
For the two-slot model of Section~sec:setup there is a finite
-tensor $D=Bbarbar ⊗⊗ L_eta(Gamma)$, with $B$ diffuse, and an
-admissible representation $delta:Gammatomathcal U(D)$calU(D)$ for each
+tensor D=Bbarbar ⊗⊗ L_eta(Gamma), with B diffuse, and an
+admissible representation delta:Gammatomathcal U(D)calU(D) for each
isolated measured factor, obtained by a fixed finite matrix corner
-equivalence. It has a sequence $gamma_n$ whose Fourier laws escape
+equivalence. It has a sequence gamma_n whose Fourier laws escape
in every measured simple factor. Rescaled continuous radial laws,
scalar effects, and complementary tightness are preserved by this
corner equivalence.
Proof.
-The complementary $Λ$-factor in the two-slot regular model is
+The complementary Λ-factor in the two-slot regular model is
diffuse; in a large enough finite matrix amplification choose a
projection in that coefficient factor with the same trace as the
-regular-model projection $e$, and identify by a partial isometry to
+regular-model projection e, and identify by a partial isometry to
get the required unital tensor-coordinate representation. Scalar
relative commutant follows from Proposition~prop:setup-induced-algebras.
@@ -4590,7 +4588,7 @@ new simple gap per stage, all target gaps diverge
unit columns and choose finite path times making the integrated mass
in successively larger Cartan windows small, with each individual error
small at the same base point (average over the source stabilizer
-interval). This gives $gamma_n$.
+interval). This gives gamma_n.
A fixed matrix corner equivalence preserves escape: approximate its
entries and adjoints in trace Hilbert norm by uniformly bounded
@@ -4598,21 +4596,21 @@ Fourier matrices, whose fixed shifts change each Cartan coordinate by
a bounded amount, so escape and rescaled radial tests persist; the
uncut Gram factor tends to the Gram of the partial isometry, and
slack preserves complementary tightness. For amplified admissibility,
-finite-rank compressions of $lambda_{X_T}(h_T)$ tend to zero when
-$r_T(h)to∞$ (embed $L^2(X_T)$ in a multiple of $L^2(T)$ and
+finite-rank compressions of lambda_{X_T}(h_T) tend to zero when
+r_T(h)to∞ (embed L^2(X_T) in a multiple of L^2(T) and
test compactly supported vectors); compress
-$Delta_T(delta(gamma_n))$ between finite-rank operators and use
-Plancherel plus escaping Fourier laws to make tracial $L^2$ norms go
+Delta_T(delta(gamma_n)) between finite-rank operators and use
+Plancherel plus escaping Fourier laws to make tracial L^2 norms go
to zero, which forbids an invariant vector.
[Entropy lower bound]
-Suppose an isolated source type $i$, with factor $S$, is matched to the
-measured type $p(i)$, with factor $T$. Let $Q_S,Q_T$ be the homogeneous
-dimensions of their boundaries. For the representation $delta$ of
+Suppose an isolated source type i, with factor S, is matched to the
+measured type p(i), with factor T. Let Q_S,Q_T be the homogeneous
+dimensions of their boundaries. For the representation delta of
Lemma~lem:scale-original-model, complementary lengths are
uniformly tight on the pure source ray, and for every
-$0<c<Q_S/Q_T$,
+0<c<Q_S/Q_T,
P_{delta,t}(t)([0,c])→0.
@@ -4625,36 +4623,36 @@ Proof.
Complementary tightness is the pure-ray conclusion of
Section~sec:transfer, transported through the corner equivalence.
Before the equivalence, complementary regularity supplies a nonzero
-constant frame column $f$ for $pi_0(Gamma)$ with $W_Λ$; if
-$e_j$ are the finitely many identity columns, the frame inequality
-applied to $W_he_j$ gives, for every $hinGamma$,
+constant frame column f for pi_0(Gamma) with W_Λ; if
+e_j are the finitely many identity columns, the frame inequality
+applied to W_he_j gives, for every hinGamma,
sum_{gammainGamma}
|D_{{h}}pi_0(gamma)f|_2^2le C_f.
-Let $Ω_t$ be a source Cartan shell of fixed positive thickness
-about length $t$ in $S$, with all compact rotations and complementary
-coordinates in a fixed compact neighborhood; $|Ω_t|asymp
-e^{Q_St}$. Restricting both section values to a compact $K$, the times
-with transporter $gamma$ lie in $Kgamma d_0(y)^{-1}$ of Haar volume
-at most $|K|$; integrating (eq:scale-bessel-count) therefore
+Let Ω_t be a source Cartan shell of fixed positive thickness
+about length t in S, with all compact rotations and complementary
+coordinates in a fixed compact neighborhood; |Ω_t|asymp
+e^{Q_St}. Restricting both section values to a compact K, the times
+with transporter gamma lie in Kgamma d_0(y)^{-1} of Haar volume
+at most |K|; integrating (eq:scale-bessel-count) therefore
bounds the averaged mass at each measured index by a constant
-independent of index and $t$.
+independent of index and t.
For a fixed complementary compact window, the number of measured
-lattice indices with $r_T(h)le c't$ is at most $Ce^{Q_Tc't}$
+lattice indices with r_T(h)le c't is at most Ce^{Q_Tc't}
(thicken on the right by a small lattice-disjoint neighborhood in the
ambient group; Cartan length changes by a bounded amount). Thus the
-restricted low-length mass averaged over $Ω_t$, divided by
-$|Ω_t|$, is at most
+restricted low-length mass averaged over Ω_t, divided by
+|Ω_t|, is at most
Ce^{(Q_Tc'-Q_S)t}.
-Choose $c<c'<Q_S/Q_T$. Lost section sets have arbitrarily small
+Choose c<c'<Q_S/Q_T. Lost section sets have arbitrarily small
measure uniformly in translating time, and the uncut translated column
has unchanged norm. Complementary tightness removes the fixed measured
window. Lemma~lem:links-invariant-variables makes every limiting
@@ -4667,106 +4665,106 @@ argument finishes.
## Tightening a Kazhdan window
-For cutoffs $L_nge1$, a bounded sequence $x_nin D_n$ is
-tight at scale $L_n$ if
+For cutoffs L_nge1, a bounded sequence x_nin D_n is
+tight at scale L_n if
lim_{Cto∞}lim_{ntomathcal U}calU
|D_{1_{{r_T>CL_n}}}x_n|_2=0
-for a free ultrafilter $mathcal U$.calU$. Ultraproducts are tracial;
+for a free ultrafilter mathcal U.calU. Ultraproducts are tracial;
ordinary subsequences are extracted once all countably many
requirements are fixed.
-The tight elements form a von Neumann subalgebra $Q_0$ of
-$prod_{mathcal U}calUD_n$, and $Q_0$ is a factor. If each $Θ_n$
-is relatively irreducible, the ultraproduct representation $Θ$
+The tight elements form a von Neumann subalgebra Q_0 of
+prod_{mathcal U}calUD_n, and Q_0 is a factor. If each Θ_n
+is relatively irreducible, the ultraproduct representation Θ
also has scalar relative commutant.
Proof.
Smooth radial truncations are uniformly bounded (Proposition~prop:ray-radial),
so tight elements admit Hilbert approximation by bounded sequences
-supported in $r_Tle CL_n$. Triangle inequality (eq:ray-triangle)
+supported in r_Tle CL_n. Triangle inequality (eq:ray-triangle)
gives closure under multiplication; length-invariance under inversion
gives closure under adjoint; tightness is closed in Hilbert norm on
each bounded ball.
-All coefficient-unitary sequences lie in $Q_0$. A central $z$
+All coefficient-unitary sequences lie in Q_0. A central z
commutes asymptotically, uniformly, with those unitaries (else choose
a violating unitary per index). The Hilbert closed convex hull of
the coefficient conjugation orbit gives a conditional expectation
-onto its commutant, so $z$ is represented, up to Hilbert-null
+onto its commutant, so z is represented, up to Hilbert-null
sequence, on the group factor alone. All fixed group labels lie in
-$Q_0$ since $L_nge1$, and conjugation by those labels acts on the
+Q_0 since L_nge1, and conjugation by those labels acts on the
orthogonal complement of the identity Fourier coefficient without
invariant vectors (ICC plus scalar phases). A fixed Kazhdan pair
-for $Gamma$ gives a uniform spectral gap even as cocycles vary.
-Approximate commutation forces $z$ scalar.
+for Gamma gives a uniform spectral gap even as cocycles vary.
+Approximate commutation forces z scalar.
## Near-identity Kazhdan tightening
[Kazhdan tightening]
-Fix a finite Kazhdan set $mathcal FcalFsubsetGamma$ and
-$ε>0$. There is $ε_K>0$ such that, if
-$Θ_n$ are admissible and
+Fix a finite Kazhdan set mathcal FcalFsubsetGamma and
+ε>0. There is ε_K>0 such that, if
+Θ_n are admissible and
|D_{1_{{r_T>L_n}}}Θ_n(gamma)|_2leε_K
- qquad(gammainmathcal F),calF),
+ (gammainmathcal F),calF),
then, after passing to a subsequence, there exist unitaries
-$u_n$ with $limsup|u_n-1|_2<ε$ such that every fixed
-element of $u_n^*Θ_n(Gamma)u_n$ is tight at scale $L_n$.
+u_n with limsup|u_n-1|_2<ε such that every fixed
+element of u_n^*Θ_n(Gamma)u_n is tight at scale L_n.
Admissibility is preserved.
Proof. [Sketch]
-Let $D_mathcal U=calU=prod_{mathcal U}calUD_n$ and $Q_0$ the tight
+Let D_mathcal U=calU=prod_{mathcal U}calUD_n and Q_0 the tight
factor of Lemma~lem:scale-tight-factor. The Jones basic
-construction $< D_mathcal U,e_calU,e_{Q_0}>$ gives
+construction < D_mathcal U,e_calU,e_{Q_0}> gives
|Θ(gamma)e_{Q_0}Θ(gamma)^*-e_{Q_0}|_2^2
=2|Θ(gamma)-E_{Q_0}Θ(gamma)|_2^2,
-which is $O(ε_K^2)$ by tightness plus smooth radial
+which is O(ε_K^2) by tightness plus smooth radial
cutoffs (Proposition~prop:ray-radial). The Hilbert-space
-Kazhdan inequality supplies a positive contraction $z$ close to
-$e_{Q_0}$; spectral rounding $p_0=1_{[1/2,1]}(z)$ gives
-$|p_0-e_{Q_0}|_2=O(ε_K)$
+Kazhdan inequality supplies a positive contraction z close to
+e_{Q_0}; spectral rounding p_0=1_{[1/2,1]}(z) gives
+|p_0-e_{Q_0}|_2=O(ε_K)
|p_0-e_{Q_0}|_2le2|z-e_{Q_0}|_2=O(ε_K).
-$p_0$ commutes with $Θ(Gamma)$ and has trace $d'$ near one;
-take a right $Q_0$-module isometry with row $Vin L^2(D_mathcal U)^calU)^{ ⊕ 2}$
-satisfying $Vq=V$, $E_{M_2(Q_0)}(V^*V)=q$, $Θ(gamma)V=Vb_gamma$;
+p_0 commutes with Θ(Gamma) and has trace d' near one;
+take a right Q_0-module isometry with row Vin L^2(D_mathcal U)^calU)^{ ⊕ 2}
+satisfying Vq=V, E_{M_2(Q_0)}(V^*V)=q, Θ(gamma)V=Vb_gamma;
VV^*=d'1.
-$e_1=(d')^{-1}V^*Vle q$ is a projection of trace one commuting
-with all $b_gamma$
+e_1=(d')^{-1}V^*Vle q is a projection of trace one commuting
+with all b_gamma
-The excess $H_1=e_1-q/d'$ is nonzero, orthogonal to $Q_0$, and
-commutes with $b_gamma$ if $d'>1$. Choose bounded representatives
-$G_n$ with exact support $r_Tge l_n$ and short approximations to
-$q,b_gamma$ with support $r_Tle a_n$:
+The excess H_1=e_1-q/d' is nonzero, orthogonal to Q_0, and
+commutes with b_gamma if d'>1. Choose bounded representatives
+G_n with exact support r_Tge l_n and short approximations to
+q,b_gamma with support r_Tle a_n:
- L_nll a_nll d_nll l_n,qquad
+ L_nll a_nll d_nll l_n,
|G_n-H_{1,n}|_2to0.
-The ray lift sends $G_n$ to $xi_n=mathbb V_bbV_{d_n}G_n$ with
+The ray lift sends G_n to xi_n=mathbb V_bbV_{d_n}G_n with
- Delta_T(b_{gamma,n})xi_n-xi_nb_{gamma,n}to0,quad
+ Delta_T(b_{gamma,n})xi_n-xi_nb_{gamma,n}to0,
Delta_T(q_n)xi_n-xi_nto0,quadxi_nq_n-xi_nto0
-and $xi'_n=Delta_T(e_{1,n})xi_n$ satisfies
+and xi'_n=Delta_T(e_{1,n})xi_n satisfies
|xi'_n|^2to(d')^{-1}|H_1|_2^2>0,
@@ -4775,7 +4773,7 @@ and $xi'_n=Delta_T(e_{1,n})xi_n$ satisfies
Delta_T(Θ'_n(gamma))xi'_n-xi'_nb_{gamma,n}to0.
-Extend $b_{gamma,n}$ off $q_n$; the normalized left vector
+Extend b_{gamma,n} off q_n; the normalized left vector
states are almost invariant in predual norm, whose square-root
densities are almost invariant vectors by Powers--Sto rmer
@@ -4784,41 +4782,41 @@ densities are almost invariant vectors by Powers--Sto rmer
Conjugation on this corner is unitarily equivalent to the
original amplified conjugation; admissibility is contradicted.
-Hence $e_1=q$, $d'=1$, and a factor equivalence gives a
-unitary $u$ with $u^*Θ(Gamma)usubset Q_0$ and
-$p_0=ue_{Q_0}u^*$. Aligning the polar part of $E_{Q_0}u$ yields
+Hence e_1=q, d'=1, and a factor equivalence gives a
+unitary u with u^*Θ(Gamma)usubset Q_0 and
+p_0=ue_{Q_0}u^*. Aligning the polar part of E_{Q_0}u yields
|u-1|_2^2le2-2|E_{Q_0}u|_2^2
=|p_0-e_{Q_0}|_2^2.
Diagonalize countably many tightness conditions; small
-$ε_K$ gives the bound.
+ε_K gives the bound.
## Composition of admissible kernels
[Exact insertion]
-Let $delta:L_alpha(Gamma)to B_0barbar ⊗⊗ L_beta(Gamma)$
+Let delta:L_alpha(Gamma)to B_0barbar ⊗⊗ L_beta(Gamma)
be a trace-preserving embedding with relatively irreducible group
representation and a sequence escaping in Fourier probability in
-every ambient simple factor. For admissible $Θ$ at an
-isolated measured factor $T$ there is an admissible
-$Θcircledastdelta$ with source multiplier $alpha$, whose
+every ambient simple factor. For admissible Θ at an
+isolated measured factor T there is an admissible
+Θcircledastdelta with source multiplier alpha, whose
spherical Fourier laws are the exact compositions of the
-kernels of $delta$ and $Θ$. The final measured cocycle
-is that of $Θ$.
+kernels of delta and Θ. The final measured cocycle
+is that of Θ.
Proof. [Sketch]
-Write $xi$ for the source multiplier of $Θ$ and put
-$T_h(y)=Θ(c(h,h^{-1}y))$. The section identity gives
-$T_g(y)T_h(g^{-1}y)=nu(g,h)(y)T_{gh}(y)$ with
-$nu(g,h)(y)=xi(c(g,g^{-1}y),c(h,g^{-1}g^{-1}y))$, a normalized
-scalar cocycle for the action on $Y$. In the twisted crossed
-product $Psi(w_h)=T_hb_h$, with multiplier $beta/nu$, is
+Write xi for the source multiplier of Θ and put
+T_h(y)=Θ(c(h,h^{-1}y)). The section identity gives
+T_g(y)T_h(g^{-1}y)=nu(g,h)(y)T_{gh}(y) with
+nu(g,h)(y)=xi(c(g,g^{-1}y),c(h,g^{-1}g^{-1}y)), a normalized
+scalar cocycle for the action on Y. In the twisted crossed
+product Psi(w_h)=T_hb_h, with multiplier beta/nu, is
trace-preserving on the twisted group algebra and extends
-normally; apply $id_{B_0} ⊗ Psi$ to $delta$. The kernel
+normally; apply id_{B_0} ⊗ Psi to delta. The kernel
formula
sum_{hinGamma}|a_{gamma,h}|_2^2
@@ -4832,7 +4830,7 @@ the induced coefficient representation
with scalar invariant space; for bounded finite tests and
-$A=sum_ia_iw_i$ one of the escaping unitaries, the coefficient
+A=sum_ia_iw_i one of the escaping unitaries, the coefficient
pairing is bounded by
sum_{i,j:,ih=h'j}|a_i|_2|a_j|_2
@@ -4841,17 +4839,17 @@ pairing is bounded by
which vanishes off the scalar summand by Fourier escape, Howe--Moore,
and the product invariant-space decomposition of Section~sec:setup.
-The amplified assertion replaces $mathcal E$calE$ by
-$L^2(Y;L^2(B(L^2(X_T))barbar ⊗⊗ D))$ and $T_g$ by
-$Delta_T(T_g)$, and admissibility of $Θ$ gives absence of
+The amplified assertion replaces mathcal EcalE by
+L^2(Y;L^2(B(L^2(X_T))barbar ⊗⊗ D)) and T_g by
+Delta_T(T_g), and admissibility of Θ gives absence of
invariant vectors.
## A linear subsequence
-Suppose $P_{Θ_n,R_n}(t_n)([a,∞])gealpha$ with
-$a,alpha>0$ and $u_n/t_nto∞$. Then
+Suppose P_{Θ_n,R_n}(t_n)([a,∞])gealpha with
+a,alpha>0 and u_n/t_nto∞. Then
P_{Θ_n,R_n}(u_n)→delta_∞,
@@ -4860,25 +4858,25 @@ uniformly under compact pre- and post-translations.
Proof.
-For fixed $m$ and large $n$, $u_n=mt_n+(u_n-mt_n)$ with
+For fixed m and large n, u_n=mt_n+(u_n-mt_n) with
nonnegative remainder. Pass to a subsequence on which the sum
laws converge; Proposition~prop:ray-addition makes the
-limit $P^{*m}*Q$ with $P([a,∞])gealpha$ and
-$Q$ supported in $[0,∞]$. For finite $M$,
+limit P^{*m}*Q with P([a,∞])gealpha and
+Q supported in [0,∞]. For finite M,
(P^{*m}*Q)([0,M])le
mathbb PbbP{Bin(m,alpha)le M/a}to0
-as $mto∞$; diagonal extraction over countably many $m$
+as mto∞; diagonal extraction over countably many m
justifies the passage. Compact changes are covered by
Lemma~lem:ray-compactshort.
[Positive mass at linear speed]
-For each isolated source type $i$, matched to $p(i)$, there
-exist $C<∞$, $alpha>0$, $T_nto∞$ such that for
-every fixed $xiinmathcal H$,calH$,
+For each isolated source type i, matched to p(i), there
+exist C<∞, alpha>0, T_nto∞ such that for
+every fixed xiinmathcal H,calH,
liminf_{n}inf_{T_n/2le tle T_n}
bigl|D_{1_{{r_{p(i)}le CT_n}}}U_{a_t}xibigr|^2
@@ -4891,298 +4889,298 @@ over moving Hilbert unit balls.
Proof. [Sketch]
-Use $delta$ from Lemma~lem:scale-original-model.
+Use delta from Lemma~lem:scale-original-model.
Suppose the matched length is totally superlinear:
P_{delta,t}(t)([0,C])to0qquad(C<∞).
Compose the original representation around the phase cycle
-$imapsto p(i)mapsto...mapsto i$ via
+imapsto p(i)mapsto...mapsto i via
Lemma~lem:scale-insertion; the first stage increases
length faster than linearly, the entropy bound prevents later
stages from losing that, and the additive structure of
Proposition~prop:ray-addition telescopes to a shorter
witness reaching the same relative threshold.
-Choose a nearly-earliest witness $(t_n,L_n,R_n=D'_nL_n)$ for
-$D'_nto∞$: feasibility is nonempty (identity embedding
-into $L_eta(Gamma)$ suffices), so the infimum
+Choose a nearly-earliest witness (t_n,L_n,R_n=D'_nL_n) for
+D'_nto∞: feasibility is nonempty (identity embedding
+into L_eta(Gamma) suffices), so the infimum
inf{tge1:|D_{1_{{r_i>L}}}Θ(gamma)|_2leε_K
(gammainmathcal F),calF), P_{Θ,D'_nL}(t)([1,∞])getfrac12}
is attained up to factor two at some witness. Lemma~lem:scale-tightening
-shrinks $Θ_n$ to a near-identity conjugate still retaining
-$1/4$ mass; $t_nto∞$; and Lemma~lem:scale-many-increments
+shrinks Θ_n to a near-identity conjugate still retaining
+1/4 mass; t_nto∞; and Lemma~lem:scale-many-increments
gives
u_n/t_nto∞⇒
P_{Θ_n,R_n}(u_n)todelta_∞.
-Insert copies of $delta$ for each cycle step. Choose
-$m_nto∞$ with
+Insert copies of delta for each cycle step. Choose
+m_nto∞ with
- t'_n=t_n/m_nto∞,quad
+ t'_n=t_n/m_nto∞,
P_{delta,t_n}(t'_n)([m_n,∞])to1,
which total superlinearity produces. Later insertions keep
-length much larger than $t_n$ by entropy; complementary
+length much larger than t_n by entropy; complementary
coordinates stay tight; the terminal kernel is tested at length
-$div t_nto∞$, so (eq:scale-minimal-escape) (with
+div t_nto∞, so (eq:scale-minimal-escape) (with
compact uniformity) sends the final length to infinity.
-A tighter feasible witness $t'_n$ contradicts $t_nle2t'_n$.
+A tighter feasible witness t'_n contradicts t_nle2t'_n.
Total superlinearity is therefore impossible. There are
-$C_0,alpha_0>0$, $T_nto∞$ with the scalar law mass
-$gealpha_0$ below $C_0T_n$, uniform for $tin[T_n/2,T_n]$
-(enlarge $C_0$ if necessary); a violating sequence plus
-Proposition~prop:ray-addition on scale $T_n$ would give
-a convolution limit for $T_n=t_n+(T_n-t_n)$, and the mass of a
-nonnegative sum below $C_0$ is at most the mass of either
+C_0,alpha_0>0, T_nto∞ with the scalar law mass
+gealpha_0 below C_0T_n, uniform for tin[T_n/2,T_n]
+(enlarge C_0 if necessary); a violating sequence plus
+Proposition~prop:ray-addition on scale T_n would give
+a convolution limit for T_n=t_n+(T_n-t_n), and the mass of a
+nonnegative sum below C_0 is at most the mass of either
summand. Lemma~lem:links-invariant-variables makes all
limiting effects scalar; Lemma~lem:scale-original-model
identifies their scalar values, and effect boundedness plus
-Hilbert density extend the bound to every fixed $xi$.
+Hilbert density extend the bound to every fixed xi.
## Horizontal differentiation and an injective height
-Fix matched isolated factors $S,T$ with boundaries $B_S,B_T$
-homogeneous dimensions $Q_S,Q_T$, Carnot distances $d_S,d_T$, compact
-angular probability $dx$, and horizontal bundles $mathcal V_R$.calV_R$. The
+Fix matched isolated factors S,T with boundaries B_S,B_T
+homogeneous dimensions Q_S,Q_T, Carnot distances d_S,d_T, compact
+angular probability dx, and horizontal bundles mathcal V_R.calV_R. The
action is horizontally conformal with horizontal dilation
-$dil(a,x)>0$, obeying
+dil(a,x)>0, obeying
dil(ab,x)=dil(a,bx)dil(b,x).
-All isolated factors here have $Q_T>2$. Let $mathscr EscrEsubsetmathcal H$calH$
+All isolated factors here have Q_T>2. Let mathscr EscrEsubsetmathcal HcalH
be the maximal invariant linear domain of
Proposition~prop:scale-pair-estimate, on which
(eq:input-10) holds uniformly over each fixed collection of
normalized mask controls.
-For $Fin C^∞(B_Ttimes B_T)$, its ordered evaluation is
-$mathcal K_calK_{x,y}(F)=iint F(w,v),P_y(dw)P_x(dv)$ (finite separated
+For Fin C^∞(B_Ttimes B_T), its ordered evaluation is
+mathcal K_calK_{x,y}(F)=iint F(w,v),P_y(dw)P_x(dv) (finite separated
decomposition on rectangles). A smooth kernel has horizontal order at
-least one if $F|_{w=v}=0$; at least two if in addition
-$d_zF|_{mathcal V_T}calV_T$ vanishes on the diagonal.
+least one if F|_{w=v}=0; at least two if in addition
+d_zF|_{mathcal V_T}calV_T vanishes on the diagonal.
Fix a compactly contained source chart, a nonnegative source cutoff
-$chi$, and a smooth nonnegative compactly supported $θ$ on the
-nilpotent group. Set $y=xdelta_ε w$ and $dnu_ε=chi(x)
-θ(w),dx,dw$. For small $ε$:
+chi, and a smooth nonnegative compactly supported θ on the
+nilpotent group. Set y=xdelta_ε w and dnu_ε=chi(x)
+θ(w),dx,dw. For small ε:
[label=(roman*)]
-item Order at least one gives, for each $xiinmathscr E$,scrE$,
-$|mathcal K_calK_{x,y}(F)xi|_{L^2(dnu_ε;mathcal H)}calH)
-le C_{F,xi}ε^{Q_S/Q_T}$.
-item If $Q_S=Q_T=Q>2$ and order is at least two, the same norm is
-$o(ε)$, explicitly bounded by
-$ε^{Q/2}$ ($2<Q<4$), $ε^2(1+|logε|)$ ($Q=4$),
-$ε^2$ ($Q>4$).
+ Order at least one gives, for each xiinmathscr E,scrE,
+|mathcal K_calK_{x,y}(F)xi|_{L^2(dnu_ε;mathcal H)}calH)
+le C_{F,xi}ε^{Q_S/Q_T}.
+ If Q_S=Q_T=Q>2 and order is at least two, the same norm is
+o(ε), explicitly bounded by
+ε^{Q/2} (2<Q<4), ε^2(1+|logε|) (Q=4),
+ε^2 (Q>4).
Proof.
-The measure pullback under $wmapsto y$ has density at most
-$Cε^{-Q_S}$ supported where $d_S(x,y)le Cε$; the pair
-measure has density $asymp d_S(x,y)^{-2Q_S}$. Thus
+The measure pullback under wmapsto y has density at most
+Cε^{-Q_S} supported where d_S(x,y)le Cε; the pair
+measure has density asymp d_S(x,y)^{-2Q_S}. Thus
int<xi,p_delta(x,y)xi>,dnu_ε
le C_xiε^{Q_S}delta^{-Q_T},
using (eq:input-10). Decompose the target-diagonal complement
-into dyadic scale-$delta$ cells $cdeltale d_T(w,v)le Cdelta$;
-such pairs form a bounded-degree graph. Order-$m$ bounds normalized
-derivatives by $Cdelta^m$. Expand in separated Fourier series, and
+into dyadic scale-delta cells cdeltale d_T(w,v)le Cdelta;
+such pairs form a bounded-degree graph. Order-m bounds normalized
+derivatives by Cdelta^m. Expand in separated Fourier series, and
absorb coefficients into masks: the annular piece is bounded in root
mean square by
C_{F,xi}delta^mmin{1,ε^{Q_S/2}delta^{-Q_T/2}},
-while the entire piece at scale $delta_0$ has
-$C_Fdelta_0^m|xi|$. Summing over $deltageε^{Q_S/Q_T}$
-for $m=1$ gives $Cε^{Q_S/Q_T}$ since $Q_T>2$. For
-$Q_S=Q_T=Q$ with $m=2$, sum at $delta_0asympε$ for the rates
+while the entire piece at scale delta_0 has
+C_Fdelta_0^m|xi|. Summing over deltageε^{Q_S/Q_T}
+for m=1 gives Cε^{Q_S/Q_T} since Q_T>2. For
+Q_S=Q_T=Q with m=2, sum at delta_0asympε for the rates
listed.
-Every matched isolated pair has $Q_S=Q_T=Q$. For every smooth
-horizontal source field $X$, every $fin C^∞(B_T)$, and every
-$xiinmathscr E$,scrE$, the distributional derivative
-$D_X(f)xi=X(xmapsto P_x(f)xi)$ lies in
-$L^2(B_S,dx;mathcal H)$,calH)$, linearly in $f,xi$ and in smooth source
-function multiples of $X$.
+Every matched isolated pair has Q_S=Q_T=Q. For every smooth
+horizontal source field X, every fin C^∞(B_T), and every
+xiinmathscr E,scrE, the distributional derivative
+D_X(f)xi=X(xmapsto P_x(f)xi) lies in
+L^2(B_S,dx;mathcal H),calH), linearly in f,xi and in smooth source
+function multiples of X.
Proof.
-For $F(w,v)=f(w)-f(v)$, ordered evaluation is $P_y(f)-P_x(f)$. A
-horizontal weight $psi_i$ with first moments the coordinate vector
+For F(w,v)=f(w)-f(v), ordered evaluation is P_y(f)-P_x(f). A
+horizontal weight psi_i with first moments the coordinate vector
gives difference quotients
-$ε^{-1}intpsi_i(w)(P_{xdelta_ε w}(f)-P_x(f))xi,dw$
-bounded in $L^2$ by $Cε^{Q_S/Q_T-1}$; adjoint pairing recovers
-the distributional derivative $-Xvarphi-(divX)varphi$.
-If $Q_S/Q_T>1$, all derivatives vanish, brackets of horizontal fields
+ε^{-1}intpsi_i(w)(P_{xdelta_ε w}(f)-P_x(f))xi,dw
+bounded in L^2 by Cε^{Q_S/Q_T-1}; adjoint pairing recovers
+the distributional derivative -Xvarphi-(divX)varphi.
+If Q_S/Q_T>1, all derivatives vanish, brackets of horizontal fields
also annihilate the same distribution, and bracket generation forces
-$P_x(f)xi$ constant on $B_S$, contradicting nonconstancy. Thus
-$Q_Sle Q_T$; cyclic permutation of the finite isolated type list
+P_x(f)xi constant on B_S, contradicting nonconstancy. Thus
+Q_Sle Q_T; cyclic permutation of the finite isolated type list
gives equality throughout. Weak compactness and the identified limit
-give $D_X(f)xiin L^2$ on charts; a partition of unity globalizes.
+give D_X(f)xiin L^2 on charts; a partition of unity globalizes.
-Set $mathcal K=L^2(B_S,dx;calK=L^2(B_S,dx;mathcal H)$calH)$ and
-$mathcal C=spancalC=span{xmapsto a(x)P_x(g)xi:
-ain L^∞(B_S), gin C^∞(B_T), xiinmathscr EscrE}$. For real
-$f$ and real horizontal $X$, the rule
+Set mathcal K=L^2(B_S,dx;calK=L^2(B_S,dx;mathcal H)calH) and
+mathcal C=spancalC=span{xmapsto a(x)P_x(g)xi:
+ain L^∞(B_S), gin C^∞(B_T), xiinmathscr EscrE}. For real
+f and real horizontal X, the rule
T_X(f)(sum_ja_jP_bullet(g_j)xi_j)
=sum_ja_jP_bullet(g_j)D_X(f)xi_j
-defines a densely defined symmetric operator on $mathcal C$.calC$. If real
-smooth $g_j,f_j$ satisfy $sum_jg_j,d_{rm hor}f_j=0$, then
+defines a densely defined symmetric operator on mathcal C.calC. If real
+smooth g_j,f_j satisfy sum_jg_j,d_{rm hor}f_j=0, then
sum_jP_bullet(g_j)T_X(f_j)=0quadhbox{on }mathcal C.calC.
Proof.
-Write $Delta_{x,y}f=P_y(f)-P_x(f)$. Sharpness gives the exact
+Write Delta_{x,y}f=P_y(f)-P_x(f). Sharpness gives the exact
identity
-$(Delta_{x,y}f)P_x(g)-P_y(g)Delta_{x,y}f
-=mathcal K_calK_{x,y}((f(w)-f(v))(g(v)-g(w)))$; the right side has
-horizontal order two and vanishes in the recovered $L^2$ average
-after dividing by $ε$. Move $Delta_{x,y}f$ by adjunction and
-replace $P_y(g)$ by $P_x(g)$ (dominated convergence); the resulting
+(Delta_{x,y}f)P_x(g)-P_y(g)Delta_{x,y}f
+=mathcal K_calK_{x,y}((f(w)-f(v))(g(v)-g(w))); the right side has
+horizontal order two and vanishes in the recovered L^2 average
+after dividing by ε. Move Delta_{x,y}f by adjunction and
+replace P_y(g) by P_x(g) (dominated convergence); the resulting
derivative of the difference quotients is
< P_x(g)D_X(f)(x)xi,eta>
=<xi,P_x(g)D_X(f)(x)eta>
-for almost every $x$. Inserting scalar parameter tests gives the
+for almost every x. Inserting scalar parameter tests gives the
pointwise form; complex weights follow by linearity. Substituting
-finitely many angular tests into a formal pairing of $u,vinmathcal C$calC$
+finitely many angular tests into a formal pairing of u,vinmathcal CcalC
proves symmetry and, when a formal expression represents zero, shows
its image is orthogonal to a dense subspace, hence zero. This gives
well-definedness, symmetry, and closability. Cancellation: the kernel
-$F(w,v)=sum_jg_j(w)(f_j(w)-f_j(v))$ has order $ge2$ when
-$sum_jg_jd_{rm hor}f_j=0$, and the same limit shows
-$sum_jP_x(g_j)D_X(f_j)(x)xi=0$; commuting further multipliers
+F(w,v)=sum_jg_j(w)(f_j(w)-f_j(v)) has order ge2 when
+sum_jg_jd_{rm hor}f_j=0, and the same limit shows
+sum_jP_x(g_j)D_X(f_j)(x)xi=0; commuting further multipliers
proves (eq:scale-horizontal-cancellation).
For a real smooth horizontal target covector field
-$theta=sum_jg_jd_{rm hor}f_j$, set $T_X(theta)=sum_j
-P_bullet(g_j)T_X(f_j)$; equation (eq:scale-horizontal-cancellation)
+theta=sum_jg_jd_{rm hor}f_j, set T_X(theta)=sum_j
+P_bullet(g_j)T_X(f_j); equation (eq:scale-horizontal-cancellation)
makes this independent of expression. (Global finitely many target
coordinate differentials span, and their Gram operator is smoothly
-invertible on $mathcal V_T$.)calV_T$.)
+invertible on mathcal V_T.)calV_T.)
-There is a positive self-adjoint measurable field $l_x$ on
-$mathcal H$,calH$, affiliated to $mathcal A_1$,calA_1$, strongly commuting with
-$P_x$ and invariant under complementary source factors, such that
+There is a positive self-adjoint measurable field l_x on
+mathcal H,calH, affiliated to mathcal A_1,calA_1, strongly commuting with
+P_x and invariant under complementary source factors, such that
- sigma_a(l_x)=dil(a,x),l_{ax},qquad
+ sigma_a(l_x)=dil(a,x),l_{ax},
beta_h(l_x)=dil(h^{-1},P_x),l_x,
-and $l$ is not identically zero.
+and l is not identically zero.
Proof.
-Choose finite smooth Parseval horizontal source fields $(X_alpha)$
-and target covectors $(theta_r)$ (invert positive Gram operators on
-$mathcal V_R$).calV_R$). Define the finite column
-$D:mathcal CcalCtomathcal K^calK^{n_Sn_T}$,
-$Du=(T_{X_alpha}(theta_r)u)_{alpha,r}$, closable by
-$mathcal C$calC$ density; set $l=|overlineoverline D|$.D|$. Parameter and angular
-unitary multipliers commute with $D$ on $mathcal C$,calC$, extending to
-the closed graph and form, so $l=int^{ ⊕ }l_x,dx$ with $l_x$
-strongly commuting with $P_x$. Unitary invariance transfers
-distributionally to $D_X(f)$ and then closes, giving affiliation and
+Choose finite smooth Parseval horizontal source fields (X_alpha)
+and target covectors (theta_r) (invert positive Gram operators on
+mathcal V_R).calV_R). Define the finite column
+D:mathcal CcalCtomathcal K^calK^{n_Sn_T},
+Du=(T_{X_alpha}(theta_r)u)_{alpha,r}, closable by
+mathcal CcalC density; set l=|overlineoverline D|.D|. Parameter and angular
+unitary multipliers commute with D on mathcal C,calC, extending to
+the closed graph and form, so l=int^{ ⊕ }l_x,dx with l_x
+strongly commuting with P_x. Unitary invariance transfers
+distributionally to D_X(f) and then closes, giving affiliation and
complementary invariance.
-For covariance, note that Parseval systems embed $mathcal V_S,calV_S,mathcal V_T$calV_T$
+For covariance, note that Parseval systems embed mathcal V_S,calV_S,mathcal V_TcalV_T
isometrically into trivial bundles whose Gram matrices are orthogonal
-projections; $Du$ lies in their ranges. Two finite Parseval systems
+projections; Du lies in their ranges. Two finite Parseval systems
thus give the same squared form on the intrinsic module core, hence the
-same $l$. Differentials change source vectors by
-$dil(a,x)$ times a horizontal isometry and target
-covectors by $dil(h^{-1},P_x)$ times an isometry, so
-(eq:scale-derivative-modulus-covariance) holds on $mathcal C$;calC$;
+same l. Differentials change source vectors by
+dil(a,x) times a horizontal isometry and target
+covectors by dil(h^{-1},P_x) times an isometry, so
+(eq:scale-derivative-modulus-covariance) holds on mathcal C;calC;
further source and target transport preserves the graph norms
(finitely many bounded dilations) and extends to the closed forms.
Uniqueness of self-adjoint operators associated with closed forms
-gives the identities. If $l=0$, then $D_X(f)xi=0$ for all smooth
-$f,X$, and the bracket argument of Lemma~lem:scale-equal-dimension
-would make $P_x$ constant, contradiction.
+gives the identities. If l=0, then D_X(f)xi=0 for all smooth
+f,X, and the bracket argument of Lemma~lem:scale-equal-dimension
+would make P_x constant, contradiction.
[Height at an isolated type]
-For matched isolated pairs $Q_S=Q_T=Q$. There are positive injective
-self-adjoint $b_x$, affiliated to $mathcal A_1$,calA_1$, strongly commuting
-with $P_x$ and invariant under complementary source factors, with
+For matched isolated pairs Q_S=Q_T=Q. There are positive injective
+self-adjoint b_x, affiliated to mathcal A_1,calA_1, strongly commuting
+with P_x and invariant under complementary source factors, with
- sigma_a(b_x)=dil(a,x)^{-Q}b_{ax},qquad
+ sigma_a(b_x)=dil(a,x)^{-Q}b_{ax},
beta_h(b_x)=dil(h^{-1},P_x)^{-Q}b_x.
The construction passes to normal copies used in amplified models.
Proof.
-Make $l_x$ everywhere covariant: at $x_0in B_S$ with parabolic
-stabilizer $J$, pull back along $gmapsto gx_0$ and set
-$F(g)=dil(g,x_0)sigma_g^{-1}(l_{gx_0})$. Haar
-disintegration over $S/J$ and
-(eq:scale-derivative-modulus-covariance) give $F(ag)=F(g)$
-almost everywhere for each fixed $ain S$; Fubini over
-$(a,g)mapsto(ag,g)$ reduces $F$ to an essential value $L_0$ (test
+Make l_x everywhere covariant: at x_0in B_S with parabolic
+stabilizer J, pull back along gmapsto gx_0 and set
+F(g)=dil(g,x_0)sigma_g^{-1}(l_{gx_0}). Haar
+disintegration over S/J and
+(eq:scale-derivative-modulus-covariance) give F(ag)=F(g)
+almost everywhere for each fixed ain S; Fubini over
+(a,g)mapsto(ag,g) reduces F to an essential value L_0 (test
via bounded resolvents and countably many matrix coefficients). For
-$kin J$, $F(gk)=dil(k,x_0)sigma_k^{-1}F(g)$, so
-$sigma_kL_0=dil(k,x_0)L_0$, defining
-$l_{gx_0}=dil(g,x_0)^{-1}sigma_gL_0$ everywhere with
-the same covariance. Target law follows by commuting $beta_h$ past
-$sigma_g$; the essential value $L_0$ satisfies the fixed-$x_0$ law
+kin J, F(gk)=dil(k,x_0)sigma_k^{-1}F(g), so
+sigma_kL_0=dil(k,x_0)L_0, defining
+l_{gx_0}=dil(g,x_0)^{-1}sigma_gL_0 everywhere with
+the same covariance. Target law follows by commuting beta_h past
+sigma_g; the essential value L_0 satisfies the fixed-x_0 law
and transports.
-Let $e_0=supp L_0$. Target dilation is bounded, positive, invertible,
-commuting with $L_0$; taking supports gives $beta_h(e_0)=e_0$, so
-$e_0inmathcal N$.calN$. Source stabilizer covariance gives
-$sigma_k(e_0)=e_0$. A split sequence in $J$ escaping in $S$ times
-sequences escaping in the complementary source factors fixes $e_0$;
+Let e_0=supp L_0. Target dilation is bounded, positive, invertible,
+commuting with L_0; taking supports gives beta_h(e_0)=e_0, so
+e_0inmathcal N.calN. Source stabilizer covariance gives
+sigma_k(e_0)=e_0. A split sequence in J escaping in S times
+sequences escaping in the complementary source factors fixes e_0;
all-factor mixing (Ssec:setup) makes translates converge to
-$tau(e_0)1$, so $e_0in{0,1}$. Nonzero column forces $e_0=1$ (this
+tau(e_0)1, so e_0in{0,1}. Nonzero column forces e_0=1 (this
uses complementary source invariance essentially).
-Every transported $l_x$ has trivial kernel; define $b_x=l_x^{-Q}$ by
+Every transported l_x has trivial kernel; define b_x=l_x^{-Q} by
spectral calculus. Dense, positive, self-adjoint, injective, affiliated
-to $mathcal A_1$;calA_1$; taking inverse powers in the covariance laws and
+to mathcal A_1;calA_1; taking inverse powers in the covariance laws and
using strong commutation with the dilation gives
(eq:scale-isolated-height-covariance).
-For amplification: on a normal copy $mathcal HcalH ⊗ mathcal L$calL$ with
-orthonormal basis $(e_n)$, an element $Xi=sum_nxi_n ⊗ e_n$ of
+For amplification: on a normal copy mathcal HcalH ⊗ mathcal LcalL with
+orthonormal basis (e_n), an element Xi=sum_nxi_n ⊗ e_n of
the maximal amplified estimate domain has every coordinate
-$xi_ninmathscr E$scrE$ (each coordinate energy is bounded by the
-total); conversely finite sums of $xi_ninmathscr E$scrE$ lie in the
+xi_ninmathscr EscrE (each coordinate energy is bounded by the
+total); conversely finite sums of xi_ninmathscr EscrE lie in the
amplified domain. Finite-coordinate projections commute with the PVMs,
preserve the domain, and commute with weak derivatives; on the
-algebraic tensor core $mathcal CcalCodotspan{e_n}$ the
-column is exactly $D ⊗ 1$, graph-densely. So
-$l^{rm amp}=l ⊗ 1$ and $b^{rm amp}=b ⊗ 1$; unitary
+algebraic tensor core mathcal CcalCodotspan{e_n} the
+column is exactly D ⊗ 1, graph-densely. So
+l^{rm amp}=l ⊗ 1 and b^{rm amp}=b ⊗ 1; unitary
identification with a normal copy preserves the identities, without
any mixing on an enlarged commutant.
The isolated branch now supplies its own-label height calculus:
-$b_x$ is positive and injective, strongly commutes with $P_x$, and
+b_x is positive and injective, strongly commutes with P_x, and
obeys (eq:scale-isolated-height-covariance) in each required
normal copy; Section~sec:heights transports its imaginary powers
through the Fourier links to establish centered length laws. The
@@ -5203,29 +5201,29 @@ Fix one measured coordinate of the exact graph relation
(eq:input-6), possibly one of the two measured coordinates of a
two-leg link. Begin with a mixed many-panel lexicographic face path that
selects every isolated gap and omits one gap in each higher-rank
-component. Let $S$ and $T$ be the noncompact simple rank-one Levi groups
+component. Let S and T be the noncompact simple rank-one Levi groups
of an omitted source position and its matched target position; compact
and ineffective factors are discarded. Source actions may be lifted to
-the simply connected cover. Write $e_i$ for the coordinate vector dual
-to the omitted simple root, so translation by $te_i$ has simple gap $t$.
+the simply connected cover. Write e_i for the coordinate vector dual
+to the omitted simple root, so translation by te_i has simple gap t.
[The real-panel upper estimate]
-There are positive constants $d_R$, determined by each real rank-one
-panel type $R$ and unchanged at the opposite type, with the following
+There are positive constants d_R, determined by each real rank-one
+panel type R and unchanged at the opposite type, with the following
property. Complete the mixed path by split-chamber translations along
its missing coordinates, taking their times to infinity successively
and more slowly than the original path. Additional slower translations
-along already selected isolated coordinates are allowed. If $t$ is the
-completing simple gap at the selected panel and $ell(g)$ the residual
+along already selected isolated coordinates are allowed. If t is the
+completing simple gap at the selected panel and ell(g) the residual
target Cartan gap, then
d_Tell(g)le d_St+O_{tight}(1).
Here tight upper error means the event where
-$d_Tell(g)-d_St$ tends to positive infinity has zero mass in the
+d_Tell(g)-d_St tends to positive infinity has zero mass in the
iterated limit, on every fixed input. One may add offsets fixed during
-preceding limits and then take further limits in those offsets; $t$ in
+preceding limits and then take further limits in those offsets; t in
(eq:input-11) is the resulting shifted gap. Gap signs are the
eventual orientations on the chosen filters.
@@ -5251,48 +5249,48 @@ calibration nor the classicality conclusion proved later is used.
The structural classification of Section~sec:setup leaves the
orthogonal, unitary, and symplectic groups as the real rank-one panel
-groups. The exceptional rank-one group $F_{4(-20)}$ cannot occur as the
+groups. The exceptional rank-one group F_{4(-20)} cannot occur as the
noncompact simple factor of a panel Levi in a higher-rank component:
its complexification would be a component of a proper absolute Dynkin
subdiagram, whereas no proper subdiagram of an irreducible Dynkin
-diagram has a component of type $F_4$. Equivalently, in the real
+diagram has a component of type F_4. Equivalently, in the real
restricted-root list a multipliable simple root arising in such a Levi
-has double-root multiplicity $1$ or $3$, and not $7$. The reduced
+has double-root multiplicity 1 or 3, and not 7. The reduced
low-rank isomorphisms lie on the orthogonal line.
## Optimal modules for a real rank-one panel
-Let $R$ be a noncompact rank-one panel factor with split generator
-$H_R$, normalized so the simple restricted gap is one. For a
-finite-dimensional complex module $V$ extended to the complexification,
+Let R be a noncompact rank-one panel factor with split generator
+H_R, normalized so the simple restricted gap is one. For a
+finite-dimensional complex module V extended to the complexification,
write
[
- V_t=ker(dpi(H_R)-t),qquad
- h(V)=sum_{t>0}dim V_t,qquad z(V)=dim V_0.
+ V_t=ker(dpi(H_R)-t),
+ h(V)=sum_{t>0}dim V_t, z(V)=dim V_0.
]
-For a nontrivial irreducible $V$, $dim V/h(V)$ is the inverse
+For a nontrivial irreducible V, dim V/h(V) is the inverse
proportion of positive-weight vectors. We extract finite-dimensional
source representations from the measured graph relation and compare
their stage dimensions with those of a target module: a sharp ratio
and a uniform gap away from the maximizers bound the dimension lost
when retaining only maximizing source summands, whose largest split
weights then bound source growth. These are the different roles of the
-constants $c_R$ and $d_R$.
+constants c_R and d_R.
We allow representations of the simply connected covering group, so
the weight lattices below are full and spin representations are
-included. Isomorphic rank-one groups are named only once: $SU(1,1)$
-uses the $SO(2,1)$ line and $Sp(1,1)$ uses the $SO(4,1)$ line, with the
+included. Isomorphic rank-one groups are named only once: SU(1,1)
+uses the SO(2,1) line and Sp(1,1) uses the SO(4,1) line, with the
reduced restricted root in both cases.
[Optimal panel modules]
-Put $k=n+1$ and let $E$ denote the standard complex module for the
+Put k=n+1 and let E denote the standard complex module for the
group in the relevant line. Among the nontrivial irreducible complex
-Lie-algebra modules, the maximum $c_R=max_Vdim V/h(V)$ and all
+Lie-algebra modules, the maximum c_R=max_Vdim V/h(V) and all
modules attaining it are
[
{ccl}
- R&c_R&maximizing modules hline
+ R&c_R&maximizing modules
SO(n,1), nge2 & k & E
SU(n,1), n>2 & k & E,E^*
SU(2,1) & 3 & E,E^*,Sym^2E,Sym^2E^*
@@ -5300,22 +5298,22 @@ modules attaining it are
Sp(2,1) & 7/2 & ∧nolimits^3_0E
]
-where $∧^3_0E$ is the primitive exterior cube (kernel of the
-symplectic contraction $∧^3Eto E$). There is
-$ε_R>0$ such that every nontrivial irreducible outside the
-table satisfies $dim V/h(V)le c_R-ε_R$. Every displayed
+where ∧^3_0E is the primitive exterior cube (kernel of the
+symplectic contraction ∧^3Eto E). There is
+ε_R>0 such that every nontrivial irreducible outside the
+table satisfies dim V/h(V)le c_R-ε_R. Every displayed
module has integral split weights and a nonzero split-weight-zero
-space, and admits a preserved nondegenerate Hermitian form $q$,
-positive definite on $V_0$, whose matrix in a compact-form Hilbert norm
-obeys $Q=Q^*=Q^{-1}$ and $dpi(H_R)^*=dpi(H_R)$. The form pairs
-$V_t$ perfectly with $V_{-t}$ and pairs it trivially with all other
-weight spaces. With $V_{>0}=⊕_{t>0}V_t$ and
-$V_{ge0}=⊕_{tge0}V_t$,
+space, and admits a preserved nondegenerate Hermitian form q,
+positive definite on V_0, whose matrix in a compact-form Hilbert norm
+obeys Q=Q^*=Q^{-1} and dpi(H_R)^*=dpi(H_R). The form pairs
+V_t perfectly with V_{-t} and pairs it trivially with all other
+weight spaces. With V_{>0}=⊕_{t>0}V_t and
+V_{ge0}=⊕_{tge0}V_t,
[
- V_{>0}^{perp_q}=V_{ge0},qquad
+ V_{>0}^{perp_q}=V_{ge0},
V_{ge0}^{perp_q}=V_{>0},
]
-and the induced form on $V_{ge0}/V_{>0}$ is positive definite. These
+and the induced form on V_{ge0}/V_{>0} is positive definite. These
two spaces are algebraic parabolic stages. The largest split weight
among the maximizing modules is
[
@@ -5326,20 +5324,20 @@ and the constants apply equally to the opposite orientation.
## Algebraic stages and compact graph operators
-Choose a target maximizing module $V$ with largest split weight $d_T$,
-$n_0=dim V$, and (if a compact diagram symmetry interchanges
-inequivalent modules) replace $V$ by the direct sum of its diagram
+Choose a target maximizing module V with largest split weight d_T,
+n_0=dim V, and (if a compact diagram symmetry interchanges
+inequivalent modules) replace V by the direct sum of its diagram
conjugates; every summand has the same optimal ratio, so still
h(V)=n_0/c_T.
Take the compact-form Hilbert norms and preserved Hermitian forms of
-Lemma~lem:panels-optimal-modules, with form symmetry $Q$, and in
-compact projectivity frames choose local or Borel lifts $pi_T(g)$ of
+Lemma~lem:panels-optimal-modules, with form symmetry Q, and in
+compact projectivity frames choose local or Borel lifts pi_T(g) of
target projectivities such that
- pi_T(g)^*Qpi_T(g)=Q,qquad
+ pi_T(g)^*Qpi_T(g)=Q,
|pi_T(g)|=exp(d_Tell(g)).