diff b34550f..b1f9bc4 arithmeticity_local_field_lattices_v2.txt
diff --git a/arithmeticity_local_field_lattices_v2.txt b/arithmeticity_local_field_lattices_v2.txt
index f57b4bd..a2453b7 100644
--- a/arithmeticity_local_field_lattices_v2.txt
+++ b/arithmeticity_local_field_lattices_v2.txt
@@ -7,10 +7,10 @@
-\usetikzlibrary{arrows.meta,positioning,calc,fit,backgrounds}
-\definecolor{linkblue}{RGB}{29,67,112}
+usetikzlibrary{arrows.meta,positioning,calc,fit,backgrounds}
+definecolor{linkblue}{RGB}{29,67,112}
-\hypersetup{pdftitle={Arithmeticity of twisted finite correspondences for lattices over local fields},pdfauthor={OpenAI},pdfsubject={Arithmetic exhaustion of twisted bifinite correspondences}}
+hypersetup{pdftitle={Arithmeticity of twisted finite correspondences for lattices over local fields},pdfauthor={OpenAI},pdfsubject={Arithmetic exhaustion of twisted bifinite correspondences}}
@@ -40,11 +40,11 @@
-\allowdisplaybreaks[2]
+allowdisplaybreaks[2]
-\title[Arithmeticity of twisted finite correspondences]{Arithmeticity of twisted finite correspondences for lattices over local fields}
+title[Arithmeticity of twisted finite correspondences]{Arithmeticity of twisted finite correspondences for lattices over local fields}
@@ -64,13 +64,13 @@ In one sentence: an analytic finite relation between the two quantum
shadows of the groups is always generated by a finite list of concrete
algebraic relations, together with their unavoidable phases.
-\maketitle
+maketitle
-\tableofcontents
-\bigskip
+tableofcontents
+bigskip
# A Guide for the Curious Reader
-\addcontentsline{toc}{section}{A Guide for the Curious Reader}
+addcontentsline{toc}{section}{A Guide for the Curious Reader}
This paper is technical, and the technical part starts in
Section~sec:intro. This first part is a guide: it explains, in
@@ -93,22 +93,22 @@ 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 $n\times n$ real matrices
-of determinant one, written $\SL_n(\mathbb R)$, is a continuous group of
-transformations of $\mathbb R^n$ 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 $mathbb R^n$bbR^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.
Inside a continuous group, one can sometimes find a lattice: a
discrete grid of symmetries that still "spreads out" through the whole
group, like the integer points
-\[
- \mathbb Z^n\subset\mathbb R^n
-\]
+[
+ mathbb Z^nbbZ^nsubsetmathbb R^nbbR^n
+]
inside the group of all translations, or the famous
-\[
- \SL_n(\mathbb Z)\subset\SL_n(\mathbb R)
-\]
+[
+ SL_n(mathbb Z)bbZ)subsetSL_n(mathbb R)bbR)
+]
of integer matrices of determinant one inside all real matrices of
determinant one. A lattice is discrete, so its elements do not
accumulate, and yet it is large enough that the continuous group is
@@ -117,11 +117,11 @@ 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 $\mathbb R$, the complex numbers $\mathbb C$,
-and the $p$-adic numbers $\mathbb Q_p$ are all "completions" of the
+theory: the real numbers $mathbb R$,bbR$, the complex numbers $mathbb C$,bbC$,
+and the $p$-adic numbers $mathbb Q_p$bbQ_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
-$\mathbb R$, $\mathbb C$, or $\mathbb Q_p$. Real place, complex place,
+$mathbb R$,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
@@ -145,11 +145,11 @@ 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
-$\mathscr K$, 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 $k_i\in\{\mathbb R,\mathbb C,finite extension of \mathbb Q_p\}$,
+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
actual isomorphism between finite-index subgroups. No irreducibility,
@@ -166,19 +166,19 @@ 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 $g\in\Gamma$ 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
+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.
@@ -186,8 +186,8 @@ 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 $\mu:\Gamma\times\Gamma\to\mathbb T$. The rule is
-\[
- \mu(g,h)\mu(gh,j)=\mu(h,j)\mu(g,hj)
-\]
-with values in the unit circle $\mathbb T$. The twisted group von
-Neumann algebra $L_\mu(\Gamma)$ is built from $\Gamma$ together with
-this cocycle: the generators $u_g$ now satisfy $u_gu_h=\mu(g,h)u_{gh}$.
+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 $u_gu_h=mu(g,h)u_{gh}$.
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
@@ -211,7 +211,7 @@ the final models.
## 5. Finite correspondences: quantized relations
-A relation between two sets $X$ and $Y$ is a subset of $X\times Y$. In
+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.
@@ -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 $A\leq\Gamma$, $B\leqΛ$ are subgroups of
-finite index, with an actual isomorphism $\delta:A\to B$ 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 $E(A,B,\delta,\sigma)$,
+[
+ sigma(a)sigma(b)=(mu(a,b))/(omega(delta(a),delta(b)))sigma(ab).
+]
+Then there is a canonical finite correspondence $E(A,B,delta,sigma)$,
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 $\Gamma\in\mathscr
-K$ and arbitrary countably infinite ICC $Λ$, is it built out of
+factors $L_mu(Gamma)$ and $L_omega(Λ)$ for $Gammainmathscr
+K$scrK$ and arbitrary countably infinite ICC $Λ$, is it built out of
arithmetic models? The answer proved here is yes.
-Main theorem (Arithmetic exhaustion). Let $\Gamma\in\mathscr K$,
-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 $A\leq\Gamma$ and
-$B\leqΛ$ have finite index, $\delta:A\to B$ 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.
@@ -273,12 +273,12 @@ 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
@@ -297,7 +297,7 @@ 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
+$Gamma$ 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.
@@ -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
@@ -345,16 +345,16 @@ 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
+item 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
+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):
+item 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
graph relation, give the analogous bound.
-\item Isolated real place (Sections~sec:ray--sec:scale):
+item 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
@@ -381,7 +381,7 @@ 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
-$\mathfrak D$, 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
the joint spectral law of the angle-height operators is exactly the law
@@ -393,7 +393,7 @@ 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
-$B_S\to B_T$ (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
after a Taylor-density argument; the gluing sections install reflection
@@ -408,7 +408,7 @@ 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 $A\to B$, 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
@@ -436,13 +436,13 @@ group is available on the comparison side at the start.
## The class, conventions, and main theorem
-Let $\mathscr K$ be the class of countably infinite ICC groups abstractly
+Let $mathscr K$scrK$ be the class of countably infinite ICC groups abstractly
commensurable with a lattice in a finite nonempty product
-\[
- G=\prod_{i=1}^r G_i,\qquad G_i=\mathbf H_i(k_i)^+.
-\]
-Here $k_i$ is $\mathbb R$, $\mathbb C$, or a finite extension of
-$\mathbb Q_p$; the primes may differ. The algebraic group $\mathbf H_i$ is
+[
+ G=prod_{i=1}^r G_i,qquad 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
@@ -458,19 +458,19 @@ hypothesis is imposed on the lattice.
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
-$\Sp(n,1)$, $n\ge2$, 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 $\mathbb T$-valued and normalized:
-\[
- \mu(g,h)\mu(gh,j)=\mu(h,j)\mu(g,hj),\qquad
- \mu(e,g)=\mu(g,e)=1.
-\]
-The twisted group factor $L_\mu(\Gamma)$ has canonical unitaries $u_g$
-with $u_gu_h=\mu(g,h)u_{gh}$ and trace $\tau(u_g)=\mathbf 1_{\{e\}}(g)$.
+All scalar cocycles are $mathbb T$-valuedbbT$-valued and normalized:
+[
+ mu(g,h)mu(gh,j)=mu(h,j)mu(g,hj),qquad
+ 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 $tau(u_g)=mathbf 1_bf1_{{e}}(g)$.
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.
@@ -482,57 +482,57 @@ 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
-\[
- Q^t=p(M_n(\mathbb C)\bar ⊗ Q)p,
- \qquad (\Tr_n ⊗ \tau_Q)(p)=t,
-\]
-with the unnormalized matrix trace $\Tr_n$; the corner itself carries its
+[
+ Q^t=p(M_n(mathbb C)bbC)barbar ⊗⊗ Q)p,
+ qquad (Tr_n ⊗ tau_Q)(p)=t,
+]
+with the unnormalized matrix trace $Tr_n$; the corner itself carries its
normalized trace.
-For finite-index subgroups $A\le\Gamma$, $B\leΛ$, a group
-isomorphism $\delta:A\to B$, 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(a)sigma(b)=(mu(a,b))/(omega(delta(a),delta(b)))sigma(ab),
+ qquad sigma(e)=I,
define the arithmetic model
- E(A,B,\delta,\sigma)
- =L^2(M)\,\bar ⊗ _{L_{\mu|_A}(A)}
- (V_\sigma ⊗ L^2(N)),
- \qquad M=L_\mu(\Gamma),\quad N=L_\omega(Λ).
+ 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).
The right $N$-action is standard. The left subgroup action on the second
-factor is $u_a(\xi ⊗ \eta)=\sigma(a)\xi ⊗ v_{\delta(a)}\eta$;
+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
-$\rho(h)\rho(k)=\omega(h,k)bar\rho(hk)$. The left action is
-$\pi(g)\xi=u_g\xi$. Hence the joint action
-$U_{(g,h)}\xi=u_g\xi v_h^*$ 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 $\Gamma\in\mathscr K$, let $Λ$ be any countably infinite ICC
-group, and let $\mu\in Z^2(\Gamma,\mathbb T)$,
-$\omega\in Z^2(Λ,\mathbb T)$ 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 $A\le\Gamma$ and $B\leΛ$ have finite index, $\delta:A\to B$
-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
-$n=\dim_{\mathbb C}V_\sigma$, the elementary model has dimensions
-\[
- \dim_{L_\mu(\Gamma)}E(A,B,\delta,\sigma)=n[Λ:B],\qquad
- \dim E(A,B,\delta,\sigma)_{L_\omega(Λ)}=n[\Gamma:A].
-\]
+$n=dim_{mathbb C}bbCV_sigma$, the elementary model has dimensions
+[
+ dim_{L_mu(Gamma)}E(A,B,delta,sigma)=n[Λ:B],qquad
+ dim E(A,B,delta,sigma)_{L_omega(Λ)}=n[Gamma:A].
+]
The theorem keeps both finite sums and closed summands, because a
@@ -545,12 +545,12 @@ 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.
@@ -561,16 +561,16 @@ 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
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,
concurrent constructions of OpenAI and Zhou give nonisomorphic ICC
property-$(T)$ groups with isomorphic group factors
-\cite[Theorem~1.2]{OpenAI2026ConnesCounterexample}
-\cite[Theorem~A]{Zhou2026Counterexamples}. The present theorem concerns
+cite[Theorem~1.2]{OpenAI2026ConnesCounterexample}
+cite[Theorem~A]{Zhou2026Counterexamples}. The present theorem concerns
the lattice class of Definition~def:class. Arithmetic exhaustion is
the finite-correspondence input used in the companion article to obtain
stable canonical recovery for this class.
@@ -589,37 +589,37 @@ Finite correspondences retain finite-index relations and their
compositions. The framework is supplied by Jones' index and module
dimensions [cite], the finite-index expectation inequality of
Pimsner--Popa [cite], and Connes' relative tensor product
-\cite[Chapter~V, Appendix~B]{Connes1994}. Vaes' classification for
+cite[Chapter~V, Appendix~B]{Connes1994}. Vaes' classification for
generalized Bernoulli factors already organizes finite-index bimodules by
subgroup isomorphisms and finite-dimensional projective representations
of cocycle ratios, and computes their fusion by projective induction
-\cite[Theorem~2.2 and Section~6.2]{Vaes2008}. Donvil--Vaes prove rigidity
+cite[Theorem~2.2 and Section~6.2]{Vaes2008}. Donvil--Vaes prove rigidity
under arbitrary scalar twists and virtual isomorphism for certain
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
+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
-cocycle is trivial on the abelian base \cite[Theorem~F]{CFOT2026}. The
+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 $\SL_n(\mathbb Z)$, $n\ge3$ \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
-\cite[Theorem~A]{Peterson2014}. Boutonnet--Houdayer developed stationary
+cite[Theorem~A]{Peterson2014}. Boutonnet--Houdayer developed stationary
and noncommutative boundary methods that recover character rigidity
-\cite[Theorems~A--C]{BoutonnetHoudayer2021}. Their noncommutative factor
+cite[Theorems~A--C]{BoutonnetHoudayer2021}. Their noncommutative factor
theorem subsequently classified intermediate von Neumann algebras of
specified lattice boundary crossed-product inclusions by parabolic
-quotients \cite[Theorem~B and Theorem~3.3]{BoutonnetHoudayer2023}.
+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
+cite[Theorem~A and Corollary~B]{HI2026}, and classify intermediate
$C^*$-algebras for boundary crossed products of irreducible lattices in
-products of real rank-one groups \cite[Corollary~B]{HoudayerLeBars2026}.
+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
measurements must be constructed in several module copies; their
@@ -628,7 +628,7 @@ simultaneous law then produces finite-rank subgroup-graph fibres.
The search for a classical equivariant map from an operator-valued
coupling is motivated in part by the self-coupling strategy in Furman's
measure-equivalence rigidity theorem
-\cite[Sections~3--4]{Furman1999}. The noncommutative classicalization and
+cite[Sections~3--4]{Furman1999}. The noncommutative classicalization and
finite-module reconstruction used below are proved here; they are not
consequences of that classical theorem.
@@ -649,23 +649,23 @@ 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 $x\in B$: $P_x(A)$ is the projection for
+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
+$Gamma$-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 $B\to B$. The two left actions
-transport its source and target labels; diagonal $\rho(Λ)$ fixes
+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
-\[
+[
[Θ_{01}(A),Θ_{02}(C)]=0
- \qquadfor all Borel sets of boundary maps A,C.
-\]
+ 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
@@ -678,8 +678,8 @@ 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
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.
@@ -703,7 +703,7 @@ finite-module spectral gap to obtain positive mass in linear windows on
fixed vectors (Sections~sec:ray--sec:scale). Pair packing
then gives horizontal derivatives, whose closed module core produces an
injective positive height with its own-label covariance. The Carnot
-boundary geometry belongs to the work of Cowling--Dooley--Kor\'anyi--Ricci
+boundary geometry belongs to the work of Cowling--Dooley--Kor'anyi--Ricci
and Pansu [cite]; the ray and finite-module
estimates used here are proved in Sections~sec:ray--sec:scale.
@@ -731,8 +731,8 @@ summand qualification in Theorem~thm:arithmetic-exhaustion
(Section~sec:recovery).
[tbp]
- \centering
- \input{figures/proof-map}
+ centering
+ input{figures/proof-map}
Caption: The geometric proof of arithmetic exhaustion. Each local branch
enters product calibration with its own speed and domain statement;
simultaneous commutation is established only afterwards.
@@ -760,94 +760,94 @@ The models of Theorem~thm:arithmetic-exhaustion admit an explicit
counting-space realization. We give the scalar transport and the
normality proof before starting the geometric converse; neither depends
on arithmetic exhaustion. Projective induction is classical
-\cite[Section~4]{Mackey1958}, and these counting-space coordinates also
+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
-\[
- D=\Gamma\timesΛ,\qquad
- \alpha((g,h),(g',h'))=\mu(g,g')\omega(h,h')bar.
-\]
-For a subgroup $C\le D$ the graph of an isomorphism
-$\delta:A\to B$ of finite-index subgroups is
-$C_\delta=\{(a,\delta(a)):a\in A\}\le D$.
+cite[Section~6.2]{Vaes2008}.
+
+Throughout, $Gamma,Λ$ are arbitrary countably infinite ICC
+groups, $M=L_mu(Gamma)$, $N=L_omega(Λ)$, and
+[
+ D=GammatimesΛ,qquad
+ 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
+$C_delta={(a,delta(a)):ain A}le D$.
## Projective induction
-Let $C\le D$ be a subgroup and let $\gamma$ be a normalized unitary
+Let $Cle D$ be a subgroup and let $gamma$ be a normalized unitary
projective representation on a finite-dimensional space $V$, with
-multiplier $\alpha|_{C\times C}$. Write $X=D/C$ for the left cosets,
-choose representatives $r_x\in D$ with $r_C=e$, and give each $x\in X$ a
+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
-$\mathcal H(C,\gamma)=⊕_{x\in X}V_x$, with the counting Hilbert
-norm. For $s\in D$ put
+$mathcal H(C,calH(C,gamma)=⊕_{xin X}V_x$, with the counting Hilbert
+norm. For $sin D$ put
- c(s,x)&=r_{sx}^{-1}sr_x,\\
+ c(s,x)&=r_{sx}^{-1}sr_x,
U^X_s(x)&=
- \frac{\alpha(s,r_x)}{\alpha(r_{sx},c(s,x))}\,
- \gamma(c(s,x)):V_x→ V_{sx}.
+ frac{alpha(s,r_x)}{alpha(r_{sx},c(s,x))},
+ gamma(c(s,x)):V_x→ V_{sx}.
[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 $A\to B$, the bundle has commuting normal $M$ and right $N$
+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
-\[
+[
c(s,tx)c(t,x)=c(st,x).
-\]
-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).
+ 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 $(z,s)(w,t)=(zw\alpha(s,t),st)$: the
-representation of the preimage of $C$ is $(z,c)\mapsto z\gamma(c)$, and
+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 $k_x\in C$. The identification from the new
+$r'_x=r_xk_x$ with $k_xin C$. The identification from the new
coordinate copy of $V_x$ to the old one is
-\[
- F_x=\alpha(r_x,k_x)^{-1}\gamma(k_x).
-\]
+[
+ 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
precisely this effect, and Equation~(eq:calculus-transport-identity)
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 $r_x^{-1}(g,e)r_x\in C$ has
+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
-\[
- \ell^2(\Gamma) ⊗ V_{x_0}→
- ⊕_{g\in\Gamma}V_{(g,e)x_0},
- \qquad \delta_g ⊗ \xi⟼ U^X_{(g,e)}\xi
-\]
-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 $\lambda_{\overline\omega}$, which are the standard normal
-representations of the right generators $\xi\mapsto\xi v_h^*$. The two
+[
+ ell^2(Gamma) ⊗ V_{x_0}→
+ ⊕_{ginGamma}V_{(g,e)x_0},
+ qquad 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
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 $\dim_{\mathbb C}V$, proving bifiniteness
+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.
@@ -858,102 +858,102 @@ the model.
[The graph-bundle unitary]
-Let $\delta:A\to B$ be a finite-index subgroup isomorphism and let
-$\sigma:A\to\mathcal U(V)$ have multiplier
-$\beta=\mu|_{A\times A}/\delta^*\omega$. Set
-$\gamma(a,\delta(a))=\sigma(a)$. Then
-\[
- E(A,B,\delta,\sigma)=
- L^2(M)\mathbin{\bar ⊗ }_{L_{\mu|A}(A)}
+Let $delta:Ato B$ be a finite-index subgroup isomorphism and let
+$sigma:Atomathcal U(V)$calU(V)$ have multiplier
+$beta=mu|_{Atimes A}/delta^*omega$. Set
+$gamma(a,delta(a))=sigma(a)$. Then
+[
+ E(A,B,delta,sigma)=
+ L^2(M)mathbin{barbar ⊗⊗ }_{L_{mu|A}(A)}
(V ⊗ L^2(N))
- \cong\mathcal H(C_\delta,\gamma).
-\]
-The unitary sends the vector $\xi$ in the fibre labelled $x$ to
-$U_{r_x}j\xi$, where
-$j\xi=\widehat1 ⊗ (\xi ⊗ \widehat1)$ in the displayed model.
+ 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.
Proof.
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 $B\backslashΛ$. On the
-summand spanned by $\xi ⊗ v_{\delta(a)}v_t$, the map
+[
+ 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
- \xi ⊗ v_{\delta(a)}v_t
- ⟼\delta_a ⊗ \sigma(a)^*\xi
- \quadin \ell^2(A) ⊗ V
+ 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
-\[
- \omega(\delta(b),\delta(a))\sigma(ba)^*\sigma(b)
- =\omega(\delta(b),\delta(a))\beta(b,a)\sigma(a)^*
- =\mu(b,a)\sigma(a)^*.
-\]
+[
+ 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
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)
- =⊕_{r\in\Gamma/A}u_r ⊗ (V ⊗ L^2(N)).
+ E(A,B,delta,sigma)
+ =⊕_{rinGamma/A}u_r ⊗ (V ⊗ L^2(N)).
At its identity fibre the graph action is exactly
-\[
- U_{(a,\delta(a))}j\xi
- =u_a ⊗ (\xi ⊗ v_{\delta(a)}^*)
- =j\sigma(a)\xi.
-\]
+[
+ U_{(a,delta(a))}jxi
+ =u_a ⊗ (xi ⊗ v_{delta(a)}^*)
+ =jsigma(a)xi.
+]
For $g=ra$, balancing in Equation~(eq:calculus-coset-model) gives
the more explicit formula
-\[
- U_{(g,h)}j\xi
- =\mu(r,a)^{-1}u_r ⊗
- (\sigma(a)\xi ⊗ v_{\delta(a)}v_h^*).
-\]
-The last Fourier coordinate is $\delta(a)h^{-1}$, up to its displayed
+[
+ 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
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
-\[
+transitive system with graph action $gamma$ at the basepoint. Finally
+[
U_sU_{r_x}j
- =\frac{\alpha(s,r_x)}{\alpha(r_{sx},c(s,x))}
- U_{r_{sx}}j\gamma(c(s,x)),
-\]
+ =frac{alpha(s,r_x)}{alpha(r_{sx},c(s,x))}
+ U_{r_{sx}}jgamma(c(s,x)),
+]
which shows that the stated unitary has exactly the transport in
Equation~(eq:input-A1).
[Elementary normality and dimensions]
-The correspondence $E(A,B,\delta,\sigma)$ is normal and bifinite. If
-$n=\dim_{\mathbb C}V_\sigma$, then
-\[
- \dim_M E(A,B,\delta,\sigma)=n[Λ:B],\qquad
- \dim E(A,B,\delta,\sigma)_N=n[\Gamma:A].
-\]
+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 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 $[\Gamma:A]$.
+convention. The respective orbit counts are $[Λ:B]$ and $[Gamma:A]$.
Finite direct sums and closed bimodule summands of these models remain
normal and bifinite: the two actions restrict normally to a reducing
subspace, and module dimension is additive and monotone under orthogonal
-projections in the commutant \cite[Theorem~10.2.1]{Jones2009}. The
+projections in the commutant cite[Theorem~10.2.1]{Jones2009}. The
geometric problem is the converse: starting from an arbitrary bifinite
correspondence, recover enough subgroup graphs to contain it. The next
sections carry out the finite-index and compact reductions needed for
@@ -973,14 +973,14 @@ 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 $L_\mu(\Gamma)$--$L_\omega(Λ)$
+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
-\[
- \rho(h)\xi=\xi v_h^*,\qquad
- \rho(h)\rho(k)=\omega(h,k)bar\rho(hk).
-\]
-The left representation is $\pi(g)\xi=u_g\xi$. Thus $\pi$ and $\rho$
+action of $L_omega(Λ)$ is written as the projective representation
+[
+ rho(h)xi=xi v_h^*,qquad
+ rho(h)rho(k)=omega(h,k)barrho(hk).
+]
+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
@@ -989,12 +989,12 @@ 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
- \Ad\rho\ on\
- L^2(\pi(L_\mu(\Gamma))',tr)
- \ominus\mathbb C1
- \quadhas no nonzero finite-dimensional subrepresentation.
+ Adrho on
+ L^2(pi(L_mu(Gamma))',tr)
+ ominusmathbb C1bbC1
+ quadhas no nonzero finite-dimensional subrepresentation.
Here the commutant is a finite factor, equipped with its normalized trace.
@@ -1004,30 +1004,30 @@ 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
+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
-\[
- \lambda_\alpha ⊗ v\simeq\lambda_{\alpha\nu} ⊗ 1_V.
-\]
+[
+ 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
-$L_{\alpha\nu}(L)$-module. Any operators commuting with every $v_h$ are
+$L_alpha(L)$-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
-\[
- T(\delta_g ⊗ \xi)=\delta_g ⊗ v_g^*\xi.
-\]
+On $ell^2(L) ⊗ V$ use the unitary
+[
+ T(delta_g ⊗ xi)=delta_g ⊗ v_g^*xi.
+]
Indeed
-\[
- T(\lambda_\alpha(h) ⊗ v_h)T^*(\delta_g ⊗ \xi)
- =\alpha(h,g)\nu(h,g)\delta_{hg} ⊗ \xi.
-\]
+[
+ T(lambda_alpha(h) ⊗ v_h)T^*(delta_g ⊗ xi)
+ =alpha(h,g)nu(h,g)delta_{hg} ⊗ xi.
+]
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
+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
unitary to such an embedding proves the second assertion. The same
formula also proves the last assertion directly.
@@ -1036,20 +1036,20 @@ 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 $Λ_0\leΛ$ such that the
+$L_mu(Gamma)$--$L_omega(Λ)$ 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
-\[
- K_c=V_c ⊗ K'_c,\qquad
- \pi(g)|_{K_c}=1 ⊗ \pi'_c(g),\qquad
- \rho(h)|_{K_c}=v^c_h ⊗ \rho'_c(h)\quad(h\inΛ_0).
-\]
+[
+ 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).
+]
The spaces $V_c$ are finite-dimensional. The $v^c_h$ are normalized
-projective matrix representations with multipliers $\nu_c$, and
+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).
+ 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
@@ -1058,65 +1058,65 @@ 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
+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,
+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 $D_c\subset P$ be the bounded elements whose $\Ad\rho$-orbits are
+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
-\[
- \|xy-x'y'\|_2\le\|x\|_∞\|y-y'\|_2+\|x-x'\|_2\|y'\|_∞,
-\]
+[
+ |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
projection onto each closed operator-norm ball is equivariant and
nonexpansive. Thus $D_c$ is a von Neumann subalgebra.
-Twisted conjugation on $L^2(N)\ominus\mathbb C1$ has no nonzero
+Twisted conjugation on $L^2(N)ominusmathbb C1$bbC1$ 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
-$E_{N^{op}}:P\to N^{op}$ is equivariant and
+$E_{N^{op}}:Pto N^{op}$ is equivariant and
$L^2$-contractive, hence $E_{N^{op}}(x)=tr(x)1$
-for $x\in D_c$. By the finite-index inequality there is $ε>0$
-with $E_{N^{op}}(x)\geε x$ for $x\ge0$, so a nonzero
-projection $q\in D_c$ has $tr(q)\geε$. A finite
+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 $zD_c=\mathcal B(V_c) ⊗ 1$. The left action
-commutes with $D_c$, hence is $1 ⊗ \pi'_c$. Choose normalized matrix
-implementers $v^c_h$ for $\Ad\rho(h)|_{zD_c}$; 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 $\mathcal B(V_c)$, giving
+$zK=V_c ⊗ K'_c$ 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
-$\overline\omega|_{Λ_0}$-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 $\nu_cbar\,\overline\omega$. The
+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
+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
+$1_{V_c} ⊗ xi$ 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).
@@ -1124,18 +1124,18 @@ is scalar. This proves Equation~(eq:setup-weak-mixing).
[Finite-index restriction and return]
-Let $\Gamma_0\le\Gamma$ and $Λ_0\leΛ$ have finite index, and
+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,
- J_L\xi&=[\Gamma:\Gamma_0]^{-1/2}
- \sum_{r\Gamma_0\in\Gamma/\Gamma_0}
- u_r ⊗ \pi(u_r)^*\xi,
- \\
- J_R\xi&=[Λ:Λ_0]^{-1/2}
- \sum_{Λ_0s\inΛ_0\backslashΛ}
- (\xi v_s^*) ⊗ v_s.
+ J_Lxi&=[Gamma:Gamma_0]^{-1/2}
+ sum_{rGamma_0inGamma/Gamma_0}
+ u_r ⊗ pi(u_r)^*xi,
+
+ J_Rxi&=[Λ:Λ_0]^{-1/2}
+ sum_{Λ_0sinΛ_0backslashΛ}
+ (xi v_s^*) ⊗ v_s.
These are bimodular isometries, with all scalar cocycles included.
@@ -1146,37 +1146,37 @@ 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
+$E_{L_{mu|Gamma_0}(Gamma_0)}(u_r^*u_{r'})=delta_{r,r'}1$; 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 $a\in\Gamma_0$, 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
+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
-$a\inΛ_0$,
-\[
- v_sv_h=(\omega(s,h))/(\omega(a,t))v_av_t,\qquad
- v_hv_t^*=(\omega(s,h))/(\omega(a,t))v_s^*v_a.
-\]
+$ainΛ_0$,
+[
+ v_sv_h=(omega(s,h))/(omega(a,t))v_av_t,qquad
+ 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
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 $\mathscr K$, 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
-finite-covolume and finite-index forms \cite[Theorem~1.7.1]{BHV2008}; see
-also \cite[Theorem~3]{DelarocheKirillov1968}. Lemmas~lem:setup-compact-reduction
+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
@@ -1185,24 +1185,24 @@ 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
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
-\[
- H=K^{ ⊗ (m+1)},\qquad
- R=\Gamma_1\times...\times\Gamma_m\timesΛ.
-\]
-The slots of $H$ are numbered $0,...,m$; $\pi_j$ denotes the left
+[
+ H=K^{ ⊗ (m+1)},qquad
+ 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 $\pi_j$, $j\ge1$, and diagonal $\rho$, with split multiplier
+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}^m\mu(g_j,g'_j)\omega(h,h')bar^{\,m+1}.
+ 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}.
@@ -1210,85 +1210,85 @@ the $\pi_j$, $j\ge1$, and diagonal $\rho$, with split multiplier
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 $\pi_j(\Gamma)$, $0\le j\le m$, and diagonal
-$\rho(Λ)$ is scalar.
+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
-$\overline\omega$-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 $\overline\omega^{\,m+1}$, 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
-$(P^{\bar ⊗ (m+1)})^{\Ad\rho^{ ⊗ (m+1)}}$, 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 $L^2(P)\ominus\mathbb C1$ gives a
+invariant tensor with a factor in $L^2(P)ominusmathbb C1$bbC1$ 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 $\mathbb C1 ⊕ (L^2(P)\ominus\mathbb C1)$ applies this to every
+as $mathbb C1bbC1 ⊕ (L^2(P)ominusmathbb C1)$bbC1)$ applies this to every
nonconstant summand.
Choose a column realization
-\[
- H=eL^2(B)^k,\qquad e\in M_k(B),\qquad
- d=(\Tr_k ⊗ \tau_B)(e)>0.
-\]
-Here $B$ is the opposite of the $\alpha_R$-twisted factor of $R$; using
+[
+ H=eL^2(B)^k,qquad ein M_k(B),qquad
+ d=(Tr_k ⊗ tau_B)(e)>0.
+]
+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
+$alpha_Rbar$-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
+multiplier $alpha_R$. Set
- Y=G/\Gamma,\qquad
- \mathcal H=L^2(Y;H),\qquad
- \mathcal N=L^∞(Y)\overline ⊗ eM_k(B)e.
+ Y=G/Gamma,qquad
+ mathcal H=L^2(Y;H),calH=L^2(Y;H),qquad
+ mathcal N=L^∞(Y)calN=L^∞(Y)overlineoverline ⊗⊗ eM_k(B)e.
-Give $Y$ its invariant probability measure. The trace on $\mathcal N$ is
-$tr_{\mathcal N}(A)
-=d^{-1}\int_Y(\Tr_k ⊗ \tau_B)(A(y))\,dy$, whereas the Hilbert norm of
+Give $Y$ its invariant probability measure. The trace on $mathcal N$calN$ 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 $d_0:Y\to G$ and write
-\[
- c(a,y)=d_0(ay)^{-1}ad_0(y),\qquad
- X_a(y)=\pi_0(c(a,y)),\qquad
+Choose a Borel section $d_0:Yto G$ and write
+[
+ c(a,y)=d_0(ay)^{-1}ad_0(y),qquad
+ X_a(y)=pi_0(c(a,y)),qquad
(U_af)(ay)=X_a(y)f(y).
-\]
+]
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 $\mathcal N$. For $1\le j\le m$, let $R'_j$ be $R$
-with $\Gamma_j$ omitted, put $\beta_j=\Ad W_{\Gamma_j}$, and let
-$\mathcal A_j$ be the commutant of the base multipliers and
+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
-\[
- (\mathcal A_j)^{\beta_j}=\mathcal N,\qquad
- (\mathcal A_j)^\sigma\ has a faithful finite $\beta_j$-invariant
- trace,\qquad
- \mathcal N^G=\mathbb C1.
-\]
+[
+ (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 N^G=calN^G=mathbb C1.bbC1.
+]
Proof.
@@ -1298,22 +1298,22 @@ $c(a,y)$ 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 $\mathcal H$. Conjugation of the corner is inner at each fibre
-and translation preserves base probability, so $\sigma$ preserves 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 $\widetilde A(g)=\pi_0(\gamma)^*A(y)\pi_0(\gamma)$.
+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 $(\mathcal A_j)^\sigma$ is the commutant of
+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 $\mathcal N^G=\mathbb C1$.
+lemma gives $mathcal N^G=calN^G=mathbb C1$.bbC1$.
## The source groups and full versus partial mixing
@@ -1329,13 +1329,13 @@ root-generated simplicity, and highest-weight theory
At a complex place the local group is regarded as a real Lie group. The
noncompact property-$(T)$ archimedean factors are the higher-rank ones and
-the groups locally isomorphic to $\Sp(n,1)$, $n\ge2$, or $F_{4(-20)}$. At
+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};
-the real rank-one classification is in \cite[Section~3.3 and
+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
@@ -1352,19 +1352,19 @@ 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
+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
-present local fields \cite[Propositions~1.12 and~1.17, Theorem~6.1]{BorelSerre1964}.
+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 $L^2(\mathcal N)$, 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 $L^2(\mathcal N)^{G_0}$,
-equivalently the conditional expectation onto $\mathcal N^{G_0}$. On the
+$G_0$, 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.
@@ -1374,42 +1374,42 @@ Proof.
Use Howe--Moore for the noncompact isotropic almost simple root-generated
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}.
+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 $\mathbf H$ over $k$, choose a nonzero parabolic unipotent
-radical $\mathbf U$. The $k$-span of the $\mathbf H(k)$-conjugates of
-$Lie(\mathbf U)$ is a nonzero invariant Lie ideal, hence all
-of $Lie(\mathbf H)$. Choose $X_1,...,X_d$ from these
+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
conjugate subspaces; the map
-$(t_1,...,t_d)\mapsto\prod_j\exp(t_jX_j)$ has invertible differential
-at zero and lands in $\mathbf H(k)^+$. 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
+cite[Proposition~1.11]{Gloeckner2008}. The same holds for the root-generated
subgroup of an algebraic simply connected cover. At an archimedean place
the generating unipotent groups are connected, so the root-generated group
is connected and open, hence the identity component; these groups are
-type~I by \cite[Theorem~1]{Dixmier1957}. At a finite place the ambient
+type~I by cite[Theorem~1]{Dixmier1957}. At a finite place the ambient
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}.
+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
-\[
- \mathcal H=\int_{\widehat S}^{ ⊕ }(H_\xi ⊗ M_\xi)\,d\eta(\xi),
- \qquad
- \pi(s)=\int_{\widehat S}^{ ⊕ }(\xi(s) ⊗ 1)\,d\eta(\xi).
-\]
+[
+ 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 $1 ⊗ V_\xi$
-\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 $V_\xi$,
-\[
- |\bigl<(\xi(s) ⊗ V_\xi)\sum_j e_j ⊗ u_j,
- \sum_k f_k ⊗ v_k\bigr>|
- \le\sum_{j,k}|<\xi(s)e_j,f_k>|\,\|u_j\|\,\|v_k\|.
-\]
+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 $V_xi$,
+[
+ |bigl<(xi(s) ⊗ V_xi)sum_j e_j ⊗ u_j,
+ sum_k f_k ⊗ v_kbigr>|
+ lesum_{j,k}|<xi(s)e_j,f_k>|,|u_j|,|v_k|.
+]
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
@@ -1418,12 +1418,12 @@ $S$-invariants.
Let $P_i$ project onto the invariants of the $i$th 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
-$L^2(\mathcal N)^{G_0}$, i.e. the trace-preserving conditional
+$L^2(mathcal N)^calN)^{G_0}$, i.e. the trace-preserving conditional
expectation.
@@ -1434,36 +1434,36 @@ 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 $\mathfrak f_{4(-20)}$: 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 $\mathbb Q_p$: 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$-ideal is either zero or the whole algebra. An adjoint
+$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 $I\subset\Pi$ has strictly positive simple
+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
-\[
- m_\alpha=\dim_{k}\mathfrak g_\alpha\ at a finite place,\qquad
- 2\rho=\sum_{\alpha>0}m_\alpha\alpha,
-\]
+$mu(g,h)$. For a positive root $alpha$ put
+[
+ m_alpha=dim_{k}mathfrak g_frakg_alpha at a finite place,qquad
+ 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 $e_i$
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.
@@ -1483,15 +1483,15 @@ split displacement.
[Haar size of Cartan shells]
For a bounded-thickness shell about a dominant vector $L$, its Haar
-measure is at most $C\exp(2\rho(L))$. For one fixed sufficiently large
-thickness there is a lower bound $c\exp(2\rho(L))$. The constants are
+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' $2\rho$.
+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 $sinh\alpha(L)$. The upper bound follows on every fixed bounded
+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
@@ -1499,13 +1499,13 @@ 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
-$[K:K\cap aKa^{-1}]$ times the fixed compact measure, comparable to
-$\exp(2\rho(L))$. 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 $\sum_{\alpha>0}m_\alpha\alpha(L)$. 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
@@ -1593,7 +1593,7 @@ denominators are bounded away from zero on a buffered larger window.
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 $t_i\in[0,1]$. The resulting point is independent of
+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
@@ -1601,12 +1601,12 @@ 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 $\alpha=\sum_in_i\alpha_i$, where
-$n_i\ge0$,
-\[
- 0\le\alpha(T_t(y-x))=\sum_in_it_iL_i\le\sum_in_iL_i=\alpha(y-x).
-\]
+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).
+]
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
@@ -1624,7 +1624,7 @@ repelling data and sufficiently large powers; the ping-pong domains give a
discrete Schottky subgroup. The full chamber action is a topologically
amenable homogeneous-space action because its minimal parabolic is
amenable; restriction to a closed subgroup preserves this amenability
-\cite[Propositions~2.2 and~2.5, Example~2.7(5)]{AnantharamanDelaroche2002}.
+cite[Propositions~2.2 and~2.5, Example~2.7(5)]{AnantharamanDelaroche2002}.
These facts are used for the source local group and its lattice, not for
any action of the comparison group on a building.
@@ -1633,8 +1633,8 @@ any action of the comparison group on a building.
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 $h\in\Gamma$ in a measured source-lattice copy; it does not
+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
@@ -1652,25 +1652,25 @@ 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: $H=eL^2(B)^k$,
-$\mathcal H=L^2(Y;H)$, $d=(\Tr ⊗ \tau_B)(e)$. For an integrable corner
+$mathcal H=L^2(Y;H)$,calH=L^2(Y;H)$, $d=(Tr ⊗ tau_B)(e)$. For an integrable corner
field write
-\[
- \mathsf T(C)=\int_Y(\Tr ⊗ \tau_B)(C(y))\,dy,\qquad
- \tau_{\mathcal N}(C)=d^{-1}\mathsf T(C).
-\]
-Thus $\tau_{\mathcal N}$ is a probability trace while the Hilbert norm of a
-column is $\|f\|_2^2=\mathsf T(ff^*)$. Matrix sizes in this section are
+[
+ mathsf T(C)=sfT(C)=int_Y(Tr ⊗ tau_B)(C(y)),dy,qquad
+ 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
fixed and finite. Operator-bounded columns form a dense subspace. The
constant corner frame is
-\[
- b_r=eε_r,\qquad 1\le r\le k,\qquad
- \sum_{r=1}^k b_rb_r^*=e.
-\]
+[
+ b_r=eε_r,qquad 1le rle k,qquad
+ 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
-$I\subset\Pi$ 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.
@@ -1679,21 +1679,21 @@ 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 $D\subset\Pi$ of diverging simple gaps is detected by rank one
+[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
between their Levi refinement spaces. Its closure has only gap sets
containing $D$.
-\item On a bounded complementary-gap window the residual blocks are
+item 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
+item 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
+item 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
@@ -1758,29 +1758,29 @@ 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
+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.
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 $V\mathcal H$
+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
-\[
- \Phi(A)=V^*E^{(1)}(A)V
-\]
+[
+ Phi(A)=V^*E^{(1)}(A)V
+]
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.
The two-column map, initially on operator-bounded columns, is
- J(\overline b,f)
- =\sqrt d\,(b(ay)^*X_a(y)f(y))^{\lim}.
+ J(overlineoverline b,f)b,f)
+ =sqrt d,(b(ay)^*X_a(y)f(y))^{lim}.
Its output has its own diagonal spectral measure $E$. The link identities
@@ -1815,19 +1815,19 @@ 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
-$T_l\to T$ in trace $L^2$. In a form tested on uncut operator-bounded
+$T_lto T$ 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
+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
another quantity tending to zero since
-$\|T_l^*T_l-T^*T\|_2\le(\|T_l\|_∞+\|T\|_∞)\|T_l-T\|_2$. 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 $Df$.
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
+$int|T(ay)-T_l(ay)|_2^2dy=|T-T_l|_2^2$, 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
@@ -1861,41 +1861,41 @@ 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 $\mathcal A_j$,
-is $\beta_j$-equivariant, and is invariant under the source face parabolic.
-Its gap-stratum masses belong to $\mathcal N$ 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
-$\mathcal Hbar ⊗ \mathcal H\to\mathcal K$. Its output forms are
+[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
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
-$N_0=\mathcal N^{G_0}$ and let $E_{N_0}:\mathcal N\to N_0$ be the
+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
trace-preserving conditional expectation. On bounded columns the link has
- \|J(\overline b,f)\|^2
- =d^2\tau_{\mathcal N}(E_{N_0}(ff^*)E_{N_0}(bb^*)).
+ |J(overlineoverline b,f)b,f)|^2
+ =d^2tau_{mathcal N}calN(E_{N_0}(ff^*)E_{N_0}(bb^*)).
Its polarized extension on finite sums,
-\[
- \Bigl\|\sum_iJ(b_ibar,f_i)\Bigr\|^2
- =d^2\sum_{i,l}\tau_{\mathcal N}
+[
+ Bigl|sum_iJ(b_ibar,f_i)Bigr|^2
+ =d^2sum_{i,l}tau_{mathcal N}calN
(E_{N_0}(f_lf_i^*)E_{N_0}(b_ib_l^*)),
-\]
+]
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,
+item For any Borel event $A$ determined by a gap stratum and its image flag,
the finite-frame identity is
- < f,\Phi(A)f>
- =\frac1d\sum_{r=1}^k\|E(A)J(b_rbar,f)\|^2.
+ < f,Phi(A)f>
+ =frac1dsum_{r=1}^k|E(A)J(b_rbar,f)|^2.
It holds on mixed and nonmixed paths, and more generally for events in
@@ -1903,7 +1903,7 @@ 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
+item 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
@@ -1916,9 +1916,9 @@ 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 $\mathcal A_j$. 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
@@ -1935,45 +1935,45 @@ 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 $\mathcal N$, 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)$, a radical-invariant vector orthogonal to the
+$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 $\mathcal N^G=\mathbb C1$.
+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
action give
-\[
- \|J(\overline b,f)\|^2
- =d\lim_a\mathsf T(ff^*\sigma_{a^{-1}}(bb^*)).
-\]
+[
+ |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
-$d^2\tau_{\mathcal N}(ff^*)\tau_{\mathcal N}(bb^*)=\|f\|_2^2\|b\|_2^2$.
+$d^2tau_{mathcal N}calN(ff^*)tau_{mathcal N}calN(bb^*)=|f|_2^2|b|_2^2$.
Polarization gives
-\[
- \Bigl\|\sum_iJ(b_ibar,f_i)\Bigr\|^2
- =d^2\sum_{i,l}\tau_{\mathcal N}(f_lf_i^*)\tau_{\mathcal N}(b_ib_l^*),
-\]
-which is the norm on $\mathcal Hbar ⊗ \mathcal H$; 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
+[
+ 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.
+
+Multiplication of an input by a start (resp. end) base function is
+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 $\mathcal Nbar$, 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 $L^2(\mathcal N)$ is the projection onto $L^2(N_0)$, which yields
+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.
@@ -1982,7 +1982,7 @@ not replace $N_0$ by scalars.
$b_r(ay)^*$ 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
+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
@@ -2004,11 +2004,11 @@ 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
+$rho(h)$ 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
+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.
@@ -2029,16 +2029,16 @@ one input.
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
+[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
+item 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,
+item 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
+item 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).
@@ -2049,35 +2049,35 @@ Proof.
Conditional expectation extends to positive $L^1$ elements. Put
$A=E_{N_0}(ff^*)$; if $A$ is bounded, (eq:links-relative-form) gives
- \|J(\overline b,f)\|^2\le d\|A\|_∞\|b\|_2^2.
+ |J(overlineoverline b,f)b,f)|^2le d|A|_∞|b|_2^2.
This proves the continuous extension from bounded $b$; the link limit is
always taken before this extension.
-Let $c\in N_0$. 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 $c\in N_0$,
-$< Qf,cQf>\le\|Q\|^2< f,cf>$; the
+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
-$\|(1-p_n)f\|_2^2=d\tau_{\mathcal N}((1-p_n)E_{N_0}(ff^*))\to0$. For
+$|(1-p_n)f|_2^2=dtau_{mathcal N}calN((1-p_n)E_{N_0}(ff^*))to0$. 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
-$n\to∞$ then $l\to∞$. The same argument with inputs reversed
+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 +2108,12 @@ 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
-$\mathcal N$; 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 $\mathcal N^G=\mathbb C1$.
+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 $\mathcal Nbar$ and $\mathcal N$; independent
+in $mathcal Nbar$calNbar$ and $mathcal N$;calN$; independent
contracted-radical invariance makes them scalar, and polarization and
normal slice separation give the scalar joint effect.
@@ -2165,9 +2165,9 @@ imposes no new mixing or scalar-commutant assertion.
Proof.
-Write the source multiplier as $\eta$. The regular representation with
-multiplier $\overline\eta$, 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 +2180,7 @@ 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 $\mathcal N$, not the enlarged commutant.
+to refer to $mathcal N$,calN$, not the enlarged commutant.
## Escape from bounded target coordinates
@@ -2194,19 +2194,19 @@ measurement and for bounded pairs in the link.
Proof.
We record the finite-module Bessel estimate used here. For a bounded column
-$b\in 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
-$\sum_{h\in R}|< bw_h^*,\xi>|^2=\|b^*\xi\|_2^2\le\|b\|_∞^2\|\xi\|_2^2$,
+$sum_{hin R}|< bw_h^*,xi>|^2=|b^*xi|_2^2le|b|_∞^2|xi|_2^2$,
projective phases being harmless. The same holds for subsets, in particular
$R'_j$-translates, and in finite matrix versions summing over fixed row and
column indices.
For bounded end and start columns, the squared norm of the coefficient at a
-fixed measured index $g\in\Gamma_j$ in $b(ay)^*X_a(y)f(y)$ is the sum over
+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
-$\Gamma_0\times R'_j$. Project each column to a finite $\Gamma_0$-coordinate
+$Gamma_0times R'_j$. 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
@@ -2216,19 +2216,19 @@ 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
+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
-$2\delta$, 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
@@ -2246,89 +2246,89 @@ 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 $\mathcal A_j$ which is target-equivariant and fixed by the
-source action $\sigma|_S$.
+space $Z$ to $mathcal A_j$calA_j$ which is target-equivariant and fixed by the
+source action $sigma|_S$.
Proof.
-Suppose $\Phi:C(Z)\to\mathcal A_j$ is such a map. Its mass $a=\Phi(1)$
-belongs to $\mathcal N^S$: target equivariance makes it commute with the
-measured target action, and membership in $\mathcal A_j$ the rest. Choose
-$ε>0$ with $p=1_{[ε,∞)}(a)\ne0$. On $p\mathcal H$,
-$\Psi(u)=pa^{-1/2}\Phi(u)a^{-1/2}p$ is unital and completely positive (a
+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 $\Phi(u)$.
+The normalization does not require $a$ to commute with $Phi(u)$.
-Choose a discrete free $F\cong\mathbb F_2\subset S$, obtained from two
+Choose a discrete free $Fcongmathbb F_2bbF_2subset S$, obtained from two
sufficiently strong rank-one contractions with disjoint attracting and
-repelling neighborhoods. Let $\mathcal D$ be the concrete $C^*$-algebra on
-$p\mathcal H$ generated by cut base multipliers, the companion actions
-$W|_{R'_j}$, and $U_f$, $f\in F$. These commute with the measured target
-action and every $\Psi$ value.
+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.
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
-$\widehat D(u ⊗ \xi)=u ⊗ D\xi$ satisfies
-$\|\widehat D\|\le\|D\|$ in the concrete input norm. The covariant lift
-$T_g(u ⊗ \xi)=(g\cdot u) ⊗ W_g\xi$ 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
-$\eta_i:Z\to\ell^2(\Gamma_j)$ with finite coordinate support, norms tending
+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
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
-$Q_i\eta=\sum_{t\in\Gamma_j}\delta_t ⊗ \pi_Z(\xi_i(\cdot)_t)\eta$;
+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,
- \Bigl\|\sum_g W_gD_g\Bigr\|
- \le\Bigl\|\sum_g\lambda_\zeta(g) ⊗ D_g\Bigr\|_{
- C^*_{r,\zeta}(\Gamma_j) ⊗ _{\min}\mathcal D}.
+ Bigl|sum_g W_gD_gBigr|
+ leBigl|sum_glambda_zeta(g) ⊗ D_gBigr|_{
+ C^*_{r,zeta}(Gamma_j) ⊗ _{min}mathcal D}calD.
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 $\Gamma_0\times R'_j$ with split multiplier. Inducing the
-$\Gamma_0$-coordinate and using $(y,h)\mapsto d_0(y)h$ turns it into the
+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
-$\sum_f\nu(f)\|\eta(f^{-1}g)\|$, and disintegrating Haar measure along left
+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
+$p$ 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
+$(R_{T_f}U_f^{(k)}p)(y)=X_f(x)p(x)cdot p(x)X_f(x)^*p(y)=p(y)$. 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
+$T_{f,n}to T_f$ in integrated tracial $L^2$ (approximate in the unit ball
by simple-base fields, then use Kaplansky density and Fourier polynomials).
-Then $S_n=\sum_f\nu(f)R_{T_{f,n}}U_f^{(k)}$ has $S_nE\to E\ne0$. 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
-\cite[Theorem~3]{Kesten1959}. The conclusions are incompatible since
-$0<\|E\|_2^2=\mathsf T(p)<∞$.
+$|S_n|le|lambda_F(nu)|$; 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)<∞$.
## Strict growth under a slower Levi translation
@@ -2347,94 +2347,94 @@ passes to strata $Ω_{D'}$ with $D'⊋ D$.
Proof.
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
+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 $D\ne\Pi$.
+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 $x:Ω_D\to X_D$ and
-minimal image dilation $V:\mathcal H\to\mathcal K_X$. The generator map
-$M_uVf\mapsto M_{u\circ x}V_Ω f$ preserves inner products and extends
-to an isometric inclusion $i:\mathcal K_X\to\mathcal K_Ω$ with
+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,
companions, and target lifts. Disintegrate
-$\mathcal K_X=\int^ ⊕ K_x\,dθ(x)$ with isometric fibre
+$mathcal K_X=calK_X=int^ ⊕ K_x,dθ(x)$ with isometric fibre
inclusions $i_x$.
-Positive density forms. Let $\mathcal F_x$ be the compact
+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
-$\omega\inΩ_D$ transports that probability to a probability
-$k_\omega(l)\,d\eta_x(l)$ on $\mathcal F_x$, positive with total mass one.
-On a compact data window $C\subsetΩ_D$, this density has bounds
-$0<c_C\le k_\omega(l)\le C_C<∞$ uniformly in $\omega\in C$ and
-$l\in\mathcal F_x$, because on such a window the residual transformations
+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
+$D_{n,x}(l)=i_x^*M_{min(k_omega(l),n)}i_x$. 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 $V\mathcal H$, so vanishes by
-minimality), then Tonelli and $\int k\,d\eta=1$ give
-\[
- \int_{X_D}\int_{\mathcal F_x}q_{x,l}(Vf_r(x))\,d\eta_x(l)dθ(x)
- =\|f_r\|^2.
-\]
+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 $\int_{\mathcal F_x}q_{x,l}(v)\,d\eta_x(l)=\|v\|^2$
-for $v\in K_x$. Consequently
+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 $\mathcal H$ to
-$\mathcal R=\int^ ⊕ _{X_D}L^2(\mathcal F_x,\eta_x;K_x)dθ(x)$.
+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
-$r_{\gamma,x}(l')=d(\gamma_*\eta_x)/d\eta_{\gamma x}(l')$. Then
-$k_{\gamma\omega}(\gamma l)=r_{\gamma,x}(\gamma l)k_\omega(l)$, and since
+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 $\mathcal R$
-includes $r_{\gamma,x}^{1/2}$ in addition to the image direct integral, so
+$D_{gamma x}(gamma l)=r_{gamma,x}(gamma l)u_{gamma,x}D_x(l)u_{gamma,x}^*$
+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
canonical lift on its minimal dilation defined by
-$U_s^X(M_uVf)=M_uVU_sf$; extend it to $\widetilde U_s$ on $\mathcal R$ 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
lifts use only the image marginal.
-The finite-trace intertwiner space. Let $\mathscr E_0$ be the bounded
-intertwiners $\mathcal H\to\mathcal R$ for the base and target actions;
-$B^*C\in\mathcal N$ for $B,C\in\mathscr E_0$, and
-$< B,C>_{\mathscr E}=\tau_{\mathcal N}(B^*C)$ completes to
-$\mathscr E$ with
+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
- \|B\|_{\mathscr E}^2=\frac1d\sum_{r=1}^k\|Bb_r\|^2,
- \qquad
- \|Bf\|^2\le d\|f\|_∞^2\|B\|_{\mathscr E}^2,
+ |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 $\mathscr U_sB=\widetilde U_sBU_s^*$ is isometric because
-$\sigma_s$ preserves $\tau_{\mathcal N}$, and base defects cancel since $B$
+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
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 $\mathscr E$; the
+$|B_n|_{op}le|B|_{op}$ converge in $mathscr E$;scrE$; the
bounded-column estimate gives strong convergence on bounded columns, and
-$B_n\to B_∞$ with $B_∞$ intertwining base and target and $S$-fixed.
-Then $u\mapsto B_∞^*M_{u(l)}B_∞$ is a nonzero positive full-flag
-map into $\mathcal A_j$, 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 $S$,
@@ -2446,29 +2446,29 @@ zero uniformly over the multiplicity unitaries; dominated convergence
finishes.
The positive compact-window comparison. For a compact window
-$C\subsetΩ_D$, 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).
+ 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
+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 $J_0f(x,l)=Vf(x)$,
-$B\mapsto< Bf,J_0f>$ is continuous on $\mathscr E$ by
+$Bmapsto< Bf,J_0f>$ is continuous on $mathscr E$scrE$ by
(eq:links-intertwiner-frame). Weak decay and the positive inequality
give
-$0\le√(c_C)< f,\Psi_s(C)f>\le< T^{(s)}f,J_0f>\to0$;
-density of bounded columns and $0\le\Psi_s(C)\le1$ 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
+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
@@ -2496,7 +2496,7 @@ if they are not.
Proof.
-Put $r=|\Pi|$ and start with a positive target stratum $D$ on a face of
+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
@@ -2504,8 +2504,8 @@ 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
@@ -2518,7 +2518,7 @@ 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 $w$
is not coaxial, the projection to a simple factor of the remaining Levi is
-nonzero, since the common kernel of $\Pi∖\{j\}$ is $\mathbb Rv$.
+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
@@ -2530,7 +2530,7 @@ 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
+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.
@@ -2541,14 +2541,14 @@ 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,
+ (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
+$A_0=sum_pa_pw_p$, $B_0=sum_qb_qw_q$, 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
+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.
@@ -2560,14 +2560,14 @@ 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
+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
+[label=(roman*)]
+item 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
+item 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.
@@ -2576,20 +2576,20 @@ 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
- \|(D_gB_0)(D_fA_0)\|_2^2
- \le C\|A_0\|_∞\|B_0\|_∞
- \|D_{h_A}A_0\|_2\|D_{h_B}B_0\|_2.
+ |(D_gB_0)(D_fA_0)|_2^2
+ le C|A_0|_∞|B_0|_∞
+ |D_{h_A}A_0|_2|D_{h_B}B_0|_2.
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
+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 $t$,
-$\|(D_gB_0)(D_fA_0)\|_2\le C\sqrt t\|A_0\|_∞\|B_0\|_∞$.
+$|(D_gB_0)(D_fA_0)|_2le Csqrt t|A_0|_∞|B_0|_∞$.
Proof.
@@ -2602,10 +2602,10 @@ 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 $t_i\in[0,1]$.
+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 $sinh(\alpha(t\cdot L))/sinh(\alpha(L))\le1$
-and the radial differential by the bounded linear map $L\mapsto t\cdot L$;
+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
@@ -2628,32 +2628,32 @@ 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
+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
+$delta=B_i'-B_i$. 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
+$ko=z+delta e_i+O(1)$ 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 $\delta$;
+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
+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
+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 $r\mapsto ur$, transverse Cartan addition plus highest-weight
-comparison gives $\|S(u)S(r)\|\ge c>0$ and $\|S(r^{-1})S(u^{-1})\|\ge c>0$
+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
@@ -2684,17 +2684,17 @@ 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
- \sum_n\sup_k|c_n(k)|<∞.
+ w_k(r)=sum_nc_n(k)h(r)u_n(r)bar,qquad
+ |u_n|_∞le1,qquad
+ sum_nsup_k|c_n(k)|<∞.
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
-$\|(\sum_nc_n(k)[A_0(D_{hu_n}A_0)^*]_k)_k\|_{\ell^2(L^2)}
-\le(\sum_n\sup_k|c_n(k)|)\|A_0\|_∞\|D_hA_0\|_2$.
+$|(sum_nc_n(k)[A_0(D_{hu_n}A_0)^*]_k)_k|_{ell^2(L^2)}
+le(sum_nsup_k|c_n(k)|)|A_0|_∞|D_hA_0|_2$.
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
@@ -2722,13 +2722,13 @@ 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
- \int_O n(v)n(v)^* ⊗ \Phi_z(dv)
+ int_O n(v)n(v)^* ⊗ Phi_z(dv)
is the full amplified input space. For the trivial module this is the
@@ -2738,7 +2738,7 @@ in the three-slot system.
Proof.
Omit $O$ first. Source covariance makes the uncut range join a projection
-over $(\mathcal A_1)^\sigma$; target covariance transports its range by
+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
finite invariant trace gives an induced range field intertwining a
@@ -2746,10 +2746,10 @@ 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
+$ran P$ 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 $A^n\ran P$.
+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
orbit of a trace-isometric action is fixed (the distance between
@@ -2778,7 +2778,7 @@ strong continuity is used only for these constructed families.
[Type comparison]
-There is a unique permutation $p$ of the full relative type list $\Pi$
+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
@@ -2813,7 +2813,7 @@ Fullness gives nonzero ordered cut product for generic source pairs
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 $j\in T_i$ and $k\in
+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.
@@ -2840,9 +2840,9 @@ 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
-$C^∞(Z)\odot\mathcal H$ bounded by both angular form norms
+$C^∞(Z)odotmathcal H$calH$ bounded by both angular form norms
- b⟼(\int_Z< b(v),\Phi_x(dv)b(v)>)^{1/2}.
+ b⟼(int_Z< b(v),Phi_x(dv)b(v)>)^{1/2}.
The generic conclusion holds throughout the transverse source orbit by
@@ -2851,8 +2851,8 @@ source covariance.
Proof.
Undo source induction: simultaneous covariance yields measurable unital
-CP maps $F_x:C(Z)\to B(H)$ with
-$F_{\gamma x}=\pi_0(\gamma)F_x\pi_0(\gamma)^*$, 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
@@ -2861,30 +2861,30 @@ 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
-$\mathfrak C$ with norm $d$. The group and matrix operators act on these
+$phi_x$ on the separable matrix-amplified full twisted crossed product
+$mathfrak C$frakC$ with norm $d$. The group and matrix operators act on these
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}).
+cite[Theorem~1.4.2]{Arveson1969}).
Fix the central support $z$ of one functional of the second family.
-$z\mathfrak C^{**}$ has separable predual, and the components $\phi_x^z$
+$zmathfrak C^frakC^{**}$ 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
-$n(x)=\sup\{\|\theta\|:0\le\theta\le\phi_x,\ \theta is z-normal\}$
+$n(x)=sup{|theta|:0lethetalephi_x, theta is z-normal}$
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.
@@ -2908,7 +2908,7 @@ 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 $P_x(U)P_y(V)\mathcal H$ for disjoint angular patches
+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,
by regular approximation, have the same spectral vector for the two
PVMs; pairing gives a common functional bounded by the two forms,
@@ -2929,11 +2929,11 @@ 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 $h_i$
+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_i\Bigl< D_{h_i(q^+)B_1}[b'(ay)^*X_a(y)f'(y)],\,
- b_i(ay)^*X_a(y)(Lf)(y)\Bigr>.
+ 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
@@ -2945,7 +2945,7 @@ 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 $h_i$ 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
by its rank-one top truncation writes the difference of the two angular
@@ -2955,11 +2955,11 @@ 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
+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 $b$
+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.
@@ -2973,99 +2973,99 @@ are equal PVMs.
## Incidence range containments
-Let $z\sim F$ mean $z$ is incident to $F$. Fix a positive stratum of a
+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
+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
-$\mathcal I_i(O)=\{x:the target face x admits incident
-type-p(i) vertex in O\}$. For an archimedean highest-line module
+$mathcal I_i(O)=calI_i(O)={x:the target face x admits incident
+type-p(i) vertex in O}$. For an archimedean highest-line module
$V$, let $R_x$ be the orthogonal projector onto
-$span\{n(v):v\sim x,\ type(v)=p(i)\}$, and
+$span{n(v):vsim x, type(v)=p(i)}$, and
define
- L_z=\ranbar\int n(v)n(v)^* ⊗ \Phi_z(dv).
+ L_z=ranbarint n(v)n(v)^* ⊗ Phi_z(dv).
[Incidence containment]
The following inclusions hold:
-[label=(\roman*)]
-\item The support space of $\Psi_F(\mathcal I_i(O))$ is contained in
-$\overline{\sum_{z\sim F}\ran\Phi_z(O)}$.
-\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
-$\overline{\sum_{z\sim F}L_z}$.
+[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
+$overline{sum_{zsim F}L_z}$.
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 $f\in\ker\Phi_z(O)$ and $b\in\ranbar\Phi_z(V_1)$, then
-$E(window)J(\bar b,f)=0$. On a mixed path $f$ may be any Hilbert
+$O$, 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
vector in the stated closed end range.
Proof.
-We prove (i) by annihilating the common kernel $\mathcal C=∩_{z\sim
-F}\ker\Phi_z(O)$. For mixed paths take any $f\in\mathcal C$ and
-operator-bounded $f_l\to f$. For nonmixed paths supported in $G_0$ with
-$N_0=\mathcal N^{G_0}$: if $i$ is in another component, ordinary fullness
-gives $\mathcal C=0$ and the inclusion is immediate; otherwise
-Lemma~lem:links-relative-boundedness(ii) makes $\mathcal C$
-$N_0$-invariant and part (iii) gives $f_l\to f$ 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 $z\sim F$.
+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 $O$,
-$z\sim F$, and an operator-bounded $b_0$. A vertex path $s^{-1}$ toward
+$zsim F$, 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
-$\ranbar\Phi_z(V_1)$. For fixed main time $a$ and index $l$,
+$ranbarPhi_z(V_1)$. 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
+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 $l\to∞$; 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
resulting end ranges. Continuity in the entire Hilbert end variable
(mixed: tensor isometry; nonmixed: Lemma~lem:links-relative-boundedness)
-gives $E(window)J(\bar b,f)=0$ for all
-$b\in\ranbar\Phi_z(V_1)$.
+gives $E(window)J(barbar b,f)=0$b,f)=0$ for all
+$binranbarPhi_z(V_1)$.
Move the end occurrence also by the independent reverse radical. As
-$z\sim F$ 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 $\mathcal I_i(O)$; monotone
-continuity gives $E(A)J(\bar b,f)=0$. Then $A$ is invariant under ignored
+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
-$c_F< f,\Psi_F(\mathcal I_i(O))f>=\frac1d\sum_r
-\|E(A)J(b_rbar,f)\|^2=0$ (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
+$h^{-1}$. 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.
+ 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
+$S(h)S(q)S(q)^*=rho(h,q)S(qh)S(q)^*$ 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
@@ -3131,10 +3131,10 @@ 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
-$W_{\widehat j}\{1,s_j\}W_{\widehat j}$. With $r$ the minimum number of
+$W_{widehatwidehat j}j{1,s_j}W_{widehatwidehat j}j$. 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 $W_{\widehat j}s_jW_{\widehat j}$. Every unequal position is above
+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.
@@ -3176,12 +3176,12 @@ 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 $i\in I$. Proposition~prop:transfer-incidence(i)
-puts $\Psi_F(\mathcal I_i(O))$ inside $P_{z_i}(O)$. A target face omitting
+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 $\Psi_F(\mathcal I_i(O)\cap\mathcal I_i(O'))$ supported in
+$O,O'$ 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 $i\in I$
+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
marginals put the joint measurement in their common multiplicative domain,
@@ -3218,7 +3218,7 @@ 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
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
@@ -3230,7 +3230,7 @@ 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 $z\mapsto sz$ under simultaneous
+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
@@ -3241,7 +3241,7 @@ projectivity from the end pencil to the start pencil.
On compact output windows the mixed link satisfies
E(dg)J(Q_zbar(dq) ⊗ P_z(du))
- \quadis supported on\quad u=gq.
+ quadis supported onquad u=gq.
The equation means annihilation by smooth joint tests vanishing on the
@@ -3335,7 +3335,7 @@ disintegrates over the product of the two endpoint spectral measures; the
strongly continuous source Levi representations commute with those PVMs
and disintegrate as well (continuous fibre representations via a countable
dense convolution algebra, standard direct integral theory
-\cite[Theorem~1.G.6]{BekkaHarpe2019}).
+cite[Theorem~1.G.6]{BekkaHarpe2019}).
Choose countable dense residue labels, compact windows, graph generators,
angular Fourier tests, fixed-level clopen tests, and total input vectors
@@ -3355,11 +3355,11 @@ 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 $\{\Delta\ne0\}$, clear a common
+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
+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
@@ -3375,28 +3375,28 @@ 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 $\widetilde S\to S$ 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 $z\mapsto sz$, 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 $\tau:B_S\to B_T$ and
+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 $z\mapsto a\tau(z)$ and $z\mapsto b\tau(z)$.
+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
- F_{z,f}(t)=(f(tP_z),f(tQ_z)),\qquad t\in T,
+ F_{z,f}(t)=(f(tP_z),f(tQ_z)),qquad tin T,
-with spectral evaluation $f(tP_z)=\int f(tu)P_z(du)$ and similarly at
+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 $f(ta\tau(z))$, with frame/projectivity
+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.
@@ -3405,28 +3405,28 @@ commutation of the endpoint families and the homogeneous form.
For a sufficiently small compact-open $U<T$, the fixed algebra
$D_*^U$ is atomic (a countable product of finite matrix algebras), and
-$D_*^T=\mathbb C1$.
+$D_*^T=mathbb C1$.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 $Ut,Ur\in U\backslash T$, jointly faithful on the
-countable set $U\backslash T$. Identify
-$\overline I ⊗ H'\congHS(I,H')$ via
-$\overline\xi ⊗ \eta\mapsto|\eta><\xi|$. For fixed
-cosets $Ut,Ur$, put $Ω_{t,r}=\{g:Utg=Ur\}$ and
-$K=J^*E(Ω_{t,r})\zeta\inHS(I,H')$. Then for every
-$d\in D_*^U$,
+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 $U\backslash
-T$ forces $D_*^T=\mathbb C1$: 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.
@@ -3449,7 +3449,7 @@ 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 $\int_U\alpha_u(d)du$ converge ultraweakly to $d$, so the ranges
+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.
@@ -3461,65 +3461,65 @@ 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 $e\in D_*^U$; shrink $V$ to fix $e$ (an atomic central block
+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
- D_0=eD_*e,\qquad D_0^U=\mathbb Ce.
+ D_0=eD_*e,qquad 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
+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 $\lambda_a\ge[U:U_n]^{-1}$).
+(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 $G_n<V\times U$ is the stabilizer of that block for
+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=\{v\in V:(v,1)\in G_n\}.
+ [K_n,K_n] acts trivially on D_n,qquad
+ 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 $v\in V$, surjectivity onto $V$ follows by
-choosing a block $b$, then a conjugate $\sigma_v(b)$ at the vacuous level.
+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.
-Let $H$ be any closed subgroup of the fixed product $V\times U$ 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 $V\times U$, such that for
-every closed subgroup $H\subset V\times U$,
+There is a constant $c$, depending only on $Vtimes U$, such that for
+every closed subgroup $Hsubset Vtimes U$,
- \exp(p^c(\Lie H\cap M_H))\subset H,
+ 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
+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 $p^c(\Lie H\cap M_H)$
+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
as in the original.
@@ -3528,28 +3528,28 @@ 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 $V_1=\{1\}$.
+open subgroup $V_1<V$ acting trivially on $D_0$, we have $V_1={1}$.
Proof.
-As $n\to∞$, $D_n$ exhausts $D_0$ ultraweakly (normal averaging and
+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 $V_1\subset V$ and $U$-compatible $D_n$. Because
+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 $[V_1,V_1]\subset\Lie V_1$ satisfies
-$[\mathfrak v_1,\mathfrak v_1]=\mathfrak v_1$ and hence $V_1$ is
+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
nontrivial, in contradiction to the hypothesis.
## Hausdorff limits and dimension equality
-The compact metrizable $V\times U$ has a compact space of closed
+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 $v\in V$, choose $(v,u_n)\in G_n$ and pass
+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.
@@ -3557,9 +3557,9 @@ Hence:
[Dimensions and abelian corner]
-$\dim_{\mathbb Q_p}\mathfrak s=\dim_{\mathbb Q_p}\mathfrak t$. Every
+$dim_{mathbb Q_p}bbQ_pmathfrak s=fraks=dim_{mathbb Q_p}bbQ_pmathfrak t$.frakt$. Every
Hausdorff limit $G_∞$ has both projections isomorphisms, and the
-target-only subgroup $G_∞\cap(1\times U)$ is $\{1\}$. Consequently
+target-only subgroup $G_∞cap(1times U)$ is ${1}$. Consequently
$D_0$ is abelian and equals $L^∞(U)$ with Haar probability.
@@ -3567,26 +3567,26 @@ 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
concatenation, adjugated with fixed determinant valuations, gives
-$p^aM_V\times\{0\}\subset(\sum_iB_{i,n}M_V)\times\{0\}\subset M_n$
+$p^aM_Vtimes{0}subset(sum_iB_{i,n}M_V)times{0}subset M_n$
uniformly at late levels. By Lemma~lem:finite-uniform-log in the
-product logarithm group, $\exp(p^{c+a}M_V)\times1\subset G_n$ uniformly.
-After enlarging $a$, $\exp(p^{c+a}M_V)$ is a fixed open $V_1\subset V$
+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
+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
-$\dim\mathfrak s=\dim\mathfrak t$). The target-only subgroup is a compact
+$Lie(G_∞)$ 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
-$\{1\}$.
+${1}$.
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
+$X_{n+1}to X_n$; 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.
@@ -3596,21 +3596,21 @@ the translation $U$-torsor with Haar probability.
[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(x\eta(s)),\qquad \eta:S\to T,
+ (beta_s f)(x)=f(xeta(s)),qquad eta:Sto T,
-where $\eta$ is a continuous homomorphism with isomorphic
-$\mathbb Q_p$-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
- Z(D_*)z_e→ eD_*e,\qquad a⟼ ae
+ Z(D_*)z_e→ eD_*e,qquad a⟼ ae
is a normal isomorphism. In the type I decomposition of this central
@@ -3621,46 +3621,46 @@ 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 $\varphi:U\to X$ be measurable onto the chart.
+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 $u_0$,
-$\varphi(vu_0)=v\varphi(u_0)$ for almost every $v$. Let
-$L=Stab_T(x_0)$. If $h\in L\cap U$, comparing $u$ and
-$uh$ on a common conull set forces $h=1$, so $L\cap U$ meets only the
+$varphi(u)=ux_0$ 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 $T/L\to X$ is a Borel bijection with Borel inverse, countably
+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 $F$,
-$g\mapsto g^{-1}F(gL)$ is left-invariant, hence constant $=nL$, giving
-$n^{-1}Ln\subset L$, equality by applying to $F^{-1}$). The injection
-$N_T(L)/L\toAut(T/L)$ has a Borel inverse. The source
+$gmapsto g^{-1}F(gL)$ 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
-$\mathbb Q_p$) homomorphism $\eta:S\to N_T(L)/L$.
+$mathbb Q_p$)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
-$\dim\mathfrak s=\dim\mathfrak t$, the closed Lie subgroup $N_T(L)$ has
-dimension equal to $T$ and is open. For $l\in L$, conjugation from the
+$dimmathfrak s=fraks=dimmathfrak t$,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.
+$L={1}$ and $Z(D_*)=L^∞(T)$ with left translation.
Remove matrix fibres: the type I decomposition and ergodicity of the
-center give $D_*\cong L^∞(T;B(\mathcal V))$ 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 $Aut(B(\mathcal V))$-
+$(alpha_ad)(x)=c(a,x)d(a^{-1}x)$ 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
the cocycle. Pure translation fixes the constant field
-$B(\mathcal V)$; Equation~(eq:finite-fixed) forces $\dim\mathcal
-V=1$.
+$B(mathcal V)$;calV)$; Equation~(eq:finite-fixed) forces $dimmathcal
+V=1$.calV=1$.
## Ordinary homogeneous panel maps
@@ -3669,16 +3669,16 @@ V=1$.
The families $(P_z)$ and $(Q_z)$ commute on their respective start and
end spaces, and there are a homogeneous analytic homeomorphism
-$\tau:B_S\to B_T$ and measurable target projectivity observations
+$tau:B_Sto B_T$ and measurable target projectivity observations
$a,b$ such that these families are evaluations of
- z⟼ a\tau(z),\qquad z⟼ b\tau(z).
+ z⟼ atau(z),qquad z⟼ btau(z).
On the independent input tensor product, the output projectivity is
sharp and equals $ab^{-1}$:
- E(W)J=J\,1_W(ab^{-1})\quadfor every Borel W\subset T.
+ E(W)J=J,1_W(ab^{-1})quadfor every Borel Wsubset T.
All statements hold coordinatewise on many-panel links.
@@ -3697,48 +3697,48 @@ 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)
+ F_0(sz)=eta(s)F_0(z)
in the ordinary measure class on $B_S$.
Regard each effective adjoint group over its coefficient field as a
-restriction of scalars to $\mathbb Q_p$; 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 $A=d\eta$ is an isomorphism of these adjoint
-algebraic groups over $\mathbb Q_p$, and
+group. Conjugation by $A=deta$ is an isomorphism of these adjoint
+algebraic groups over $mathbb Q_p$,bbQ_p$, and
- A\Ad(s)=\Ad(\eta(s))A,\qquad s\in S.
+ AAd(s)=Ad(eta(s))A,qquad 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 $\tau:B_S\to B_T$ exists, intertwining $S$ with
-$\eta(S)$.
+original field); thus $tau:B_Sto B_T$ exists, intertwining $S$ with
+$eta(S)$.
-Write $B_S=S/P$; the measurable lift $s\mapsto\eta(s)^{-1}F_0(sP)$ is
+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
+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
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
+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
-$a\tau(z)=g\,b\tau(z)$ for countably dense labels, and since $\tau$ is
+$atau(z)=g,btau(z)$ 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
@@ -3750,30 +3750,30 @@ 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 $\mathbb Q_p$, and
-let $\ell_R$ be its rank-one Cartan gap. Constants $d_R>0$ depend only
+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_R\ell_R(r)+O(1),\qquad r\in R,
+ log|Ad(r)|=d_Rell_R(r)+O(1),qquad 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 $|\cdot|_p^{[k:\mathbb Q_p]}$
-with the largest relative root multiple $r_{\max}\in\{1,2\}$,
-$d_R=r_{\max}/[k:\mathbb Q_p]$. 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
from a single type list.
[Uniform finite-place upper speed]
-After the inner limit of a translated panel link, let $t\ge0$ 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
- d_T\ell_T(g)\le d_St+O_{tight}(1),
+ d_Tell_T(g)le d_St+O_{tight}(1),
with tightness on each fixed ordinary input vector, uniformly in the
@@ -3785,11 +3785,11 @@ 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 $s_t\in S$ and $\eta(s_t)$ is
+($t$ 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 $a\tau(s_tz)=g\,b\tau(z)$, so
+Proposition~prop:finite-maps give $atau(s_tz)=g,btau(z)$, so
- g=a\eta(s_t)b^{-1},
+ g=aeta(s_t)b^{-1},
where $s_t$ is deterministic with total missing gap $t$. Crucially
@@ -3798,16 +3798,16 @@ 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 $A=d\eta$, Equation~(eq:finite-adjoint-intertwining) gives
-$\Ad(\eta(s_t))=A\Ad(s_t)A^{-1}$, hence
-$\log\|\Ad(\eta(s_t))\|\le d_St+C$, and the same for $s_t^{-1}$. Fix
-$\xi\in\overline I ⊗ H'$ 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 $d_T\ell_T(g)\le d_St+C_{\xi,ε}$
+(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.
@@ -3848,28 +3848,28 @@ measure.
## Coordinates and a uniform smooth calculus
-The horospherical group is $N_b=V\times Z$ with
-\[
- (v,z)(v',z')=(v+v',z+z'+\tfrac12[v,v']),\qquad
- \delta_R(v,z)=(Rv,R^2z).
-\]
+The horospherical group is $N_b=Vtimes Z$ with
+[
+ (v,z)(v',z')=(v+v',z+z'+tfrac12[v,v']),qquad
+ delta_R(v,z)=(Rv,R^2z).
+]
Choose inner products with $< J_zv,w>=< z,[v,w]>$
-and $J_z^2=-|z|^2\id$; the horizontal layer is bracket-generating. Set
-$Q=\dim V+2\dim Z$. In half-space coordinates $(n,\lambda)\in
-N_b\times(0,∞)$, with a fixed normalization of Haar measure,
+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=(d\lambda^2)/(\lambda^2)
- +(|dv|^2)/(4\lambda^2)
- +(|dz-\tfrac12[v,dv]|^2)/(4\lambda^4),
- \qquad dp=\frac{dn\,d\lambda}{\lambda^{Q+1}}.
+ 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}}.
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
(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
+$d_{∂ X}$ 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
@@ -3879,99 +3879,99 @@ 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 $m\ge0$:
-[label=(\roman*)]
-\item If $(v,z)=\log(n^{-1}n')$, then
+norms for every $mge0$:
+[label=(roman*)]
+item 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)}
- {2\lambda\lambda'}.
+ cosh d((n,lambda),(n',lambda'))
+ =frac{√((lambda^2+lambda'^2+|v|^2/4)^2+|z|^2)}
+ {2lambdalambda'}.
-\item Changes between the compactly rotated horospherical boundary charts
-preserve normalized bounds on boxes of every scale $0<R\le1$, and in the
-adjoined height $0\le\lambda\le CR$. The conversion between the outward
+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
- \lambda=e^{-r}b,
+ n=n_0(e^{-2r}U,e^{-4r}W),qquad
+ lambda=e^{-r}b,
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
+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 $q'=e^{-L+u}$, $0\le u\le L$, with normalized derivatives
-bounded. A smaller boundary box of radius $\delta\le cq$, height at most
-$c\delta$, has image radius at most $Ce^{2u-L}\delta$. Exterior pieces
-$q\ge c$ have output upper bound $Ce^{-L}$.
+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
-$A=\lambda^2+1+|v|^2/4$, $C=A^2+|z|^2$, $c=\sqrt C/(2\lambda)$. In the
-orthonormal frame the three components of $\nabla c$ are
-$\lambda A/\sqrt C-\sqrt C/(2\lambda)$, $(Av+J_zv)/(2\sqrt C)$,
-$\lambda z/\sqrt C$. Since $J_zv\perp v$ and $|J_zv|^2=|z|^2|v|^2$, the
+$A=lambda^2+1+|v|^2/4$, $C=A^2+|z|^2$, $c=sqrt C/(2lambda)$. 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},
- \frac\lambda{√(B_0)}),\quad
- A_0=\lambda^2+|v|^2/4,\quad B_0=A_0^2+|z|^2,
+ 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,
-with $j\delta_R=\delta_{R^{-1}}j$. On a compact annulus
-$(|v|^4/16+|z|^2)^{1/4}+\lambda\asymp1$, 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 $a_{\log\lambda}$ 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
+(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 $T∂ X/\mathcal V$
-induced by the surjective bracket $∧^2\mathcal V\to
-T∂ X/\mathcal V$. It is compact-invariant with mass one. Naturality
+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
of the bracket makes it the compact angular probability, and its Jacobian
-under a horizontal dilation $d\cdotIsom$ is exactly $d^Q$.
+under a horizontal dilation $dcdotIsom$ is exactly $d^Q$.
[Triangle defect]
-There is $C$ such that if $x,y\in X$ and
+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 $d_{∂ X}(g^-,h^+)\le Ce^{-b/2}$.
+$d_{∂ X}(x^+,y^+)le Ce^{-b/2}$. 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
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 $d(x,y)\ge d(o,x)+d(o,y)+2\log\delta-C$. Apply to $x=g^{-1}o$,
+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:
-[label=(\roman*)]
-\item A smooth symbol on the product of two fixed buffered boxes with
+[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
+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
+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
same bound.
@@ -3991,48 +3991,48 @@ by the factorization.
## Uniform radial multipliers
[Radial cutoffs]
-Let $f\in C^∞([0,∞))$ 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 $R\ge R_f$. Consequently for any
-homomorphism $\Gamma\toIsom(X)$, 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
-\[
- D_{f(r/R)}:\ B\,\bar ⊗ L_\eta(\Gamma)\to
- B\,\bar ⊗ L_\eta(\Gamma),\qquad
- \sum_hY_hu_h⟼\sum_hf(r(h)/R)Y_hu_h
-\]
+[
+ D_{f(r/R)}: B,barbar ⊗⊗ L_eta(Gamma)to
+ B,barbar ⊗⊗ L_eta(Gamma),qquad
+ 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.
+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 $s=-\log\lambda$,
-$s'=-\log\lambda'$. Since $|s-s'|\le d(x,x')$, only a bounded number of
+$[0,2a]$ 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 $|s|,|s'|\le C_bR$.
+dilation reduces to $|s|,|s'|le C_bR$.
-Take smooth radial cutoffs $\chi_j(n^{-1}n')$ at radii
+Take smooth radial cutoffs $chi_j(n^{-1}n')$ at radii
$ε_j=e^{-j}$, equal to one on a smaller ball, and
-$J=\lceil C'_bR\rceil$. The annuli $\chi_j-\chi_{j+1}$ for
-$-J\le j<J$ and the innermost ball $\chi_J$ cover the support of
+$J=lceil C'_bRrceil$. 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
-\[
- H_j=\log(\max(ε_j,\lambda,\lambda')^2)/(\lambda\lambda').
-\]
+[
+ 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 $j,R$.
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
smooth pieces. The telescoping identity
-\[
- \sum_{j=-J}^{J-1}\phi_j(\chi_j-\chi_{j+1})+\phi_J\chi_J
- =\phi_{-J}\chi_{-J}+\sum_{j=-J+1}^J(\phi_j-\phi_{j-1})\chi_j
-\]
-with $\phi_j=f(H_j/R)$ and $\phi_{-J}=0$ finishes: each ball kernel has
+[
+ 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
uniformly bounded Schur norm by smooth separation, and there are
$O(R)$ pieces of size $O(R^{-1})$ each.
@@ -4044,45 +4044,45 @@ unchanged, giving complete boundedness.
## One-sided localization
-Let $D=B\bar ⊗ L_\eta(\Gamma)$ with product trace $\tau$ and
-Fourier coefficients $Y_h\in L^2(B)$; $D_\varphi$ denotes coefficient
+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,
-$(YY^*)_g=\sum_h\eta(gh,h^{-1})\overline{\eta(h,h^{-1})}Y_{gh}Y_h^*$,
+$(YY^*)_g=sum_heta(gh,h^{-1})overline{eta(h,h^{-1})}Y_{gh}Y_h^*$,
and phases have absolute value one.
[Unary Plancherel estimate]
-Let $\varphi_{i,\nu}:\Gamma\to\mathbb C$ satisfy
-$\sup_h\sum_i|\varphi_{i,\nu}(h)|^2\le b$ for every $\nu$. With $T_g$ an
+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
+$z_{g,t}=sum_nu c_{g,t,nu}[Y(D_{varphi_{i(g,t),nu}}Y)^*]_g$ where
+$|c_{g,t,nu}|le Aa_nu$, $sum_nu a_nu<∞$, we have
- (\sum_g\sum_{t\in T_g}\|z_{g,t}\|_{2,B}^2)^{1/2}
- \le bA(\sum_\nu a_\nu)\|Y\|_∞\|Y\|_2.
+ (sum_gsum_{tin T_g}|z_{g,t}|_{2,B}^2)^{1/2}
+ le bA(sum_nu a_nu)|Y|_∞|Y|_2.
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 $\delta=e^{-s}$, $s\ge1$. Let $(f_C)_C$ be smooth masks
-supported on buffered boundary boxes of radius $\delta$, with fixed
-overlap and normalized $C^m$ bound. If $Y\in D$ has Fourier support in
-$\{r\ge l\}$ with $l\ge Cs+C$, then with $A_C=D_{f_C(h^+)}Y$,
+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_C\Bigr\|_2^2
- \le C\|Y\|_∞^2\|Y\|_2^2.
+ sum_{C,D'}|A_CA_{D'}^*|_2^2
+ =Bigl|sum_CA_C^*A_CBigr|_2^2
+ le C|Y|_∞^2|Y|_2^2.
The constants may depend on the mask bound but not on the number of
masks, and the assertion holds with arbitrary coefficient algebras and
@@ -4091,12 +4091,12 @@ scalar twists.
Proof.
Estimate the coefficients of $A_CA_{D'}^*$ at fixed shift $g$; put
-$L=r(g)$; the weights in $YY^*$ are $f_C((gh)^+)\overline{f_{D'}(h^+)}$.
+$L=r(g)$; 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 $2u\le L+C_1$ (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
@@ -4104,34 +4104,34 @@ $2u>L+C_1$ (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 $C\delta$, giving the
+buffered $delta$-box into a ball of radius $Cdelta$, giving the
required multiplicity.
-The exceptional pairs lie within $C_2\delta$ 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
- L\ge 2s-C_3.
+ 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 $e^{-u}$,
-$0\le u\le L/2+C_4$, and on such an annulus the product seminorm obeys
+$0le ule L/2+C_4$, and on such an annulus the product seminorm obeys
- C_m\min(1,e^{s-u})\min(1,e^{s-L+u})\le C'_m e^{-|s-u|}.
+ 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
boundedly many such boxes meet the annulus. Lemmas~lem:ray-cells
-and~lem:ray-unary give $Ce^{-|s-u|}\|Y\|_∞\|Y\|_2$ for the
+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 $\{r\ge l\}$ and zero on
-$\{r\le l/2\}$; Proposition~prop:ray-radial bounds the approximants
+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
@@ -4143,103 +4143,103 @@ finishes.
## The ray lift and its densities
-Let $\mathcal K_X=L^2(X,dp)$ and let $T_g$ be translation on
-$\mathcal K_X$. Define
-\[
- \Delta:D→ B(\mathcal K_X)\bar ⊗ D,\qquad
- \Delta(bu_h)=T_h ⊗ bu_h,
-\]
+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$. Define
+[
+ Delta:D→ B(mathcal K_X)calK_X)barbar ⊗⊗ D,qquad
+ Delta(bu_h)=T_h ⊗ bu_h,
+]
a normal unital homomorphism (conjugate by the unitary sending
-$\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 $\supp\psi\Subset(1,2)$, $F$
+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 $d_1\ge1$, basepoint $x$, and
+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)).
+ 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
-$\mathbb V_{d_1}Y=\sum_h\theta_{o,d_1}(\cdot,h^+) ⊗ Y_hu_h$.
+$mathbb V_bbV_{d_1}Y=sum_htheta_{o,d_1}(cdot,h^+) ⊗ Y_hu_h$.
-For a column $\xi\in\mathcal K_X ⊗ L^2(D)$, the left and right
-densities $\rho^L_\xi,\rho^R_\xi\in L^1(D)$ 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
-$d_1\to∞$, $l/d_1\to∞$, and $Y\in D$ Fourier-supported in
-$\{r\ge l\}$.
-[label=(\roman*)]
-\item For large indices, $\mathbb V_{d_1}Y$ 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
-$\|\mathbb V_{d_1}Y(\mathbb V_{d_1}Y)^*\|_{2,\Tr ⊗ \tau}$ has this
+$d_1to∞$, $l/d_1to∞$, 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
bound.
-\item If $\supp\widehat x\subset\{r\le a\}$ and $a/d_1\to0$, then
-$\|\mathbb V_{d_1}(xY)-\Delta(x)\mathbb V_{d_1}Y\|_2\to0$ and
-$\|\mathbb V_{d_1}(Yx)-(\mathbb V_{d_1}Y)x\|_2\to0$.
-\item For another long column $Z$ and bounded $x$, if for every
-contributing triple $h\in\supp\widehat Y$, $hk\in\supp\widehat Z$,
-$k\in\supp\widehat x$ one has
-$d_{∂ X}(h^+,(hk)^+)\le e^{-cl'}$ with $c>0$,
-$l'/d_1\to∞$, then
-
- <\mathbb V_{d_1}Z,(\mathbb V_{d_1}Y)x>
+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
+
+ <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 $K_{d_1}(v,w)=\int\theta_{o,d_1}(p,v)\theta_{o,d_1}(p,w)\,dp$
-and disintegrate at radial depth $s=d(o,p)\in[d_1-C,2d_1+C]$. With
+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 $n_p=n_x\delta_{\lambda_p}q$,
-$t=\log(\lambda_p/\lambda_x)/d_1$ fixes the normalization; polar volume
-$J(s)\asymp e^{Qs}$ compensates the angular shadow $O(e^{-Qs})$, and
+$d_1,s$. 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 $e^{-s}$.
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
+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
+hypotheses of Lemma~lem:ray-angular. Tracial H"older plus
+(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
-$\|\mathbb V_{d_1}Y(\mathbb V_{d_1}Y)^*\|_2^2
-=\|(\mathbb V_{d_1}Y)^*\mathbb V_{d_1}Y\|_2^2$ 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
-$w_{d_1}(g,v)=\int_X\theta_{o,d_1}(p,v)\theta_{go,d_1}(p,v)\,dp$.
+$w_{d_1}(g,v)=int_Xtheta_{o,d_1}(p,v)theta_{go,d_1}(p,v),dp$.
There are smooth annular partitions about $g^+$ at integer depths
-$0\le j\le L$ (with exterior piece and innermost ball) on which
+$0le jle L$ (with exterior piece and innermost ball) on which
- \|w_{d_1}(g,\cdot)-a_{d_1}(j,L)\|_{C^m_{\rm normalized}}
- &\le C_m/d_1,\\
- a_{d_1}(j,L)&=\frac1{d_1}\int_{s>j}\psi(s/d_1)
- \psi((s-2j+L)/d_1)\,ds,
+ |w_{d_1}(g,cdot)-a_{d_1}(j,L)|_{C^m_{rm normalized}}
+ &le C_m/d_1,
+ a_{d_1}(j,L)&=frac1{d_1}int_{s>j}psi(s/d_1)
+ psi((s-2j+L)/d_1),ds,
-with constants uniform in $L,d_1\ge1$, and
+with constants uniform in $L,d_1ge1$, and
- |∂_ja_{d_1}(j,L)|\le C/d_1,\qquad
- a_{d_1}(0,L)=1+O(\min(1,L/d_1)).
+ |∂_ja_{d_1}(j,L)|le C/d_1,qquad
+ a_{d_1}(0,L)=1+O(min(1,L/d_1)).
@@ -4248,96 +4248,96 @@ 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
+$delta_{√(D_1)}L_{P_1^{-1}}jL_{v^{-1}}$ 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
- e^\beta=ε√(D_1/D_ε).
+ n=delta_{√(D_1)}(P_1^{-1}P_ε),qquad
+ 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=\tfrac12\log(\Delta_R/D_t),\quad
- \delta_Rn=m(R,t,u)=\delta_{√(\Delta_R)}(Q_R^{-1}P_t).
+ beta=2j-L+B(R,t,u),quad
+ B=tfrac12log(Delta_R/D_t),quad
+ 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
-$m\to P_0(u)\ne e$ as $R,t\to0$. The rest follows by writing
-$H(m)=\int F(q)F(m^{-1}q)\,dq$ (smooth, compact, $H(e)=1$):
+$mto P_0(u)ne e$ 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)=\int\psi(t)\psi(t-\beta/d_1)H(\delta_{e^{-td_1}}n)\,dt.
+ 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
+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 $s\le j$. The $O(1/d_1)$ derivative bounds follow similarly.
+$H$ 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
+$(gh)^+,gh^+$; 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,
+ 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
+$le4d_1+C$ (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
+$O(d_1^{-1})|Y|_∞|Y|_2$ 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)
gives the bound. The stronger short-coefficient version
$a=o(d_1)$ gives difference bounded in $L^2$ by
- C((1+a)/d_1+e^{C'd_1-l})\|Y\|_∞\|Y\|_2.
+ C((1+a)/d_1+e^{C'd_1-l})|Y|_∞|Y|_2.
Polarization extends to
- <\mathbb V_{d_1}Z,\Delta(x)\mathbb V_{d_1}Y>
+ <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
-$\supp\widehat x\subset\{r\le a\}$. Bounded approximants and
+$suppwidehatwidehat xxsubset{rle a}$. 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
- \|K_{d_1}-1\|_{C^m_{\rm normalized}}\le C_m e^{C_m d_1-d_2},
+ |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 $d_1\ll d_2\ll l'$. Separate $K_{d_1}-1$ at
+$e^{-2d_2}$. 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
+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
-$\|\mathbb V_{d_1}(Yx)-(\mathbb V_{d_1}Y)x\|_2^2$ finishes (ii) for
+$|mathbb V_bbV_{d_1}(Yx)-(mathbb V_bbV_{d_1}Y)x|_2^2$ 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
-$\|\Delta(x-x')\mathbb V_{d_1}Y\|_2\to0$ and
-$\|(\mathbb V_{d_1}Y)(x-x')\|_2\to0$; the analogous amplified
+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.
@@ -4345,11 +4345,11 @@ Proof.
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_{d_1}Y\|_2^2,
- \|(\mathbb V_{d_1}Y)z\|_2^2\}
- \le C\|Y\|_∞\|Y\|_2\|z\|_∞\|z\|_2.
+ 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 $\mathbb V_{d_1}Y(\mathbb V_{d_1}Y)^*$
+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.
@@ -4363,54 +4363,54 @@ with symbol $f$. Finite factor traces are normalized.
An admissible representation for $T$ is a finite factor
-$D=B\bar ⊗ L_\eta(\Gamma)$ together with a normalized
-projective representation $Θ:\Gamma\to\mathcal U(D)$ such that
-$D\capΘ(\Gamma)'=\mathbb C1$ and the representation
+$D=Bbarbar ⊗⊗ L_eta(Gamma)$ together with a normalized
+projective representation $Θ:Gammatomathcal U(D)$calU(D)$ such that
+$DcapΘ(Gamma)'=mathbb C1$bbC1$ and the representation
- \gamma⟼\Ad\Delta_T(Θ(\gamma))
- \quad\hbox{on}\quad
- L^2(B(L^2(X_T))\bar ⊗ D,\Tr ⊗ \tau)
+ gamma⟼AdDelta_T(Θ(gamma))
+ quadhbox{on}quad
+ 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 $a\in G$ 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
- P_{Θ,R}(t)(E)=\int_Y
- \bigl\|D_{1_{\{r_T/R\in E\}}}X^Θ_{a_t}(y)\bigr\|_2^2\,dy,
+ 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 $E\subset[0,∞]$.
+for Borel $Esubset[0,∞]$.
[Uniform short approximation]
-Let $R_n\to∞$ 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)\|_2\to0
- \quad(\gamma\in\Gamma,\ ε>0).
+ |D_{1_{{r_T>ε R_n}}}Θ_n(gamma)|_2to0
+ quad(gammainGamma, ε>0).
Then there are $a_n=o(R_n)$ and uniformly operator-bounded fields
-$Z_{n,s}$ with Fourier support in $\{r_T\le a_n\}$ such that
+$Z_{n,s}$ with Fourier support in ${r_Tle a_n}$ such that
- \sup_{s\in C}\|X^{Θ_n}_s-Z_{n,s}\|_{L^2(Y;L^2(D_n))}\to0
+ sup_{sin C}|X^{Θ_n}_s-Z_{n,s}|_{L^2(Y;L^2(D_n))}to0
-for every compact $C\subset G$. 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
-$c(s,y)\in\Gamma\cap K^{-1}CK$, a finite set; apply
+$2ε$ 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
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),qquad |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
@@ -4427,22 +4427,22 @@ 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 $V\cup CV$ and their inverses, for a compact neighborhood
+$G$; a symmetric compactly supported density $nu$ that is bounded
+below on $Vcup CV$ and their inverses, for a compact neighborhood
$V$ of $1$, satisfies
-$\sup_{g\in C}\|\pi(g)\xi-\xi\|^2\le
-C_1\int\|\pi(h)\xi-\xi\|^2d\nu(h)$, 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 $\chi$
+$sup_{gin C}|pi(g)xi-xi|^2le
+C_1int|pi(h)xi-xi|^2dnu(h)$, 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 $R_n\to∞$, $Θ_n$ 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 $t_n,u_n\ge0$, if $P_{Θ_n,R_n}(t_n)$
+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.
@@ -4462,15 +4462,15 @@ sum.
Replacing the actual transporter by its short approximation, average
$s$ over $m_0^{*k}$. For bounded smooth radial cuts
$A_{n,f}=D_{f(r_T/R_n)}X^{Θ_n}_{a_{t_n}}$,
-$B_{n,g}=D_{g(r_T/R_n)}X^{Θ_n}_{a_{u_n}}(a_{u_n}^{-1}\cdot)$,
+$B_{n,g}=D_{g(r_T/R_n)}X^{Θ_n}_{a_{u_n}}(a_{u_n}^{-1}cdot)$,
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,\qquad
- f ⊗ g⟼(A_{n,f}(sy)X^{Θ_n}_s(y)B_{n,g}(y))^{\lim}.
+ J:L^2(P_t) ⊗ L^2(P_u)→mathcal K,calK,qquad
+ 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
@@ -4478,20 +4478,20 @@ 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
- \tau(\zeta_n(sy)^*\Delta_T(X^{Θ_n}_s(y))\xi_n(y)C(y,s))
+ tau(zeta_n(sy)^*Delta_T(X^{Θ_n}_s(y))xi_n(y)C(y,s))
-with $\xi_n=\mathbb V_{d_n}B$, $\zeta_n=\mathbb V_{d_n}(A^*)$. 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
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.
+$|z|_∞|z|_2|rho^L_{xi_n}|_2$, i.e.
- \|\Delta_T(z)\xi_n\|^2
- \le\|z\|_∞\|z\|_2\|\rho^L_{\xi_n}\|_2.
+ |Delta_T(z)xi_n|^2
+ le|z|_∞|z|_2|rho^L_{xi_n}|_2.
-Limit $n\to∞$ then $k\to∞$.
+Limit $nto∞$ then $kto∞$.
For input values $0$ and $+∞$, use triangle inequalities and
short approximations with smooth tail cutoffs (Proposition~prop:ray-radial);
@@ -4501,14 +4501,14 @@ 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
+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 $\varphi\in C([0,∞])$.
+$E(varphi)J=JM_{varphi(s+t)}$ for $varphiin C([0,∞])$.
Convergence on the whole sequence follows by compactness of
probability measures on $[0,∞]$.
@@ -4517,11 +4517,11 @@ probability measures on $[0,∞]$.
At an isolated type, Section~sec:transfer has already supplied
-sharp boundary measurements $x\mapsto P_x$. 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
Jacobians. The construction takes an inverse power of the modulus of a
-horizontal derivative of $x\mapsto P_x$; 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,
@@ -4535,7 +4535,7 @@ 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 $\Sp(n,1)$, $n\ge2$, 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 +4543,7 @@ 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
@@ -4567,10 +4567,10 @@ not new assumptions.
For the two-slot model of Section~sec:setup there is a finite
-tensor $D=B\bar ⊗ L_\eta(\Gamma)$, with $B$ diffuse, and an
-admissible representation $\delta:\Gamma\to\mathcal U(D)$ 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.
@@ -4590,7 +4590,7 @@ 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
@@ -4598,10 +4598,10 @@ 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
+$Delta_T(delta(gamma_n))$ 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.
@@ -4609,12 +4609,12 @@ 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
+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
$0<c<Q_S/Q_T$,
- P_{\delta,t}(t)([0,c])→0.
+ P_{delta,t}(t)([0,c])→0.
Both hold uniformly under source pre- and post-translations in
@@ -4625,26 +4625,26 @@ 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
+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 $h\in\Gamma$,
+applied to $W_he_j$ gives, for every $hinGamma$,
- \sum_{\gamma\in\Gamma}
- \|D_{\{h\}}\pi_0(\gamma)f\|_2^2\le C_f.
+ 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
+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 $K\gamma d_0(y)^{-1}$ of Haar volume
+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 $t$.
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
@@ -4667,20 +4667,20 @@ argument finishes.
## Tightening a Kazhdan window
-For cutoffs $L_n\ge1$, a bounded sequence $x_n\in D_n$ is
+For cutoffs $L_nge1$, a bounded sequence $x_nin D_n$ is
tight at scale $L_n$ if
- \lim_{C\to∞}\lim_{n\to\mathcal U}
- \|D_{1_{\{r_T>CL_n\}}}x_n\|_2=0
+ lim_{Cto∞}lim_{ntomathcal U}calU
+ |D_{1_{{r_T>CL_n}}}x_n|_2=0
-for a free ultrafilter $\mathcal U$. 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
-$\prod_{\mathcal U}D_n$, and $Q_0$ is a factor. If each $Θ_n$
+$prod_{mathcal U}calUD_n$, and $Q_0$ is a factor. If each $Θ_n$
is relatively irreducible, the ultraproduct representation $Θ$
also has scalar relative commutant.
@@ -4688,7 +4688,7 @@ 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 $r_T\le CL_n$. 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.
@@ -4699,97 +4699,97 @@ 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
sequence, on the group factor alone. All fixed group labels lie in
-$Q_0$ since $L_n\ge1$, 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.
+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 $\mathcal F\subset\Gamma$ 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)\|_2\leε_K
- \qquad(\gamma\in\mathcal F),
+ |D_{1_{{r_T>L_n}}}Θ_n(gamma)|_2leε_K
+ qquad(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 $D_\mathcal U=\prod_{\mathcal U}D_n$ 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 $< D_\mathcal U,e_{Q_0}>$ 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,
+ |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
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
-$\|p_0-e_{Q_0}\|_2=O(ε_K)$
+$|p_0-e_{Q_0}|_2=O(ε_K)$
- \|p_0-e_{Q_0}\|_2\le2\|z-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 $V\in L^2(D_\mathcal U)^{ ⊕ 2}$
-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.
-$e_1=(d')^{-1}V^*V\le q$ 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 $r_T\ge l_n$ and short approximations to
-$q,b_\gamma$ with support $r_T\le a_n$:
+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_n\ll a_n\ll d_n\ll l_n,\qquad
- \|G_n-H_{1,n}\|_2\to0.
+ L_nll a_nll d_nll l_n,qquad
+ |G_n-H_{1,n}|_2to0.
-The ray lift sends $G_n$ to $\xi_n=\mathbb V_{d_n}G_n$ 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(q_n)\xi_n-\xi_n\to0,\quad\xi_nq_n-\xi_n\to0
+ Delta_T(b_{gamma,n})xi_n-xi_nb_{gamma,n}to0,quad
+ 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\|^2\to(d')^{-1}\|H_1\|_2^2>0,
+ |xi'_n|^2to(d')^{-1}|H_1|_2^2>0,
- \Delta_T(Θ'_n(\gamma))\xi'_n-\xi'_nb_{\gamma,n}\to0.
+ 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--St\o rmer
+densities are almost invariant vectors by Powers--Sto rmer
- \|A^{1/2}-B^{1/2}\|_2^2\le\|A-B\|_1\quad(A,B\ge0).
+ |A^{1/2}-B^{1/2}|_2^2le|A-B|_1quad(A,Bge0).
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 $u^*Θ(\Gamma)u\subset Q_0$ and
+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\|_2^2\le2-2\|E_{Q_0}u\|_2^2
- =\|p_0-e_{Q_0}\|_2^2.
+ |u-1|_2^2le2-2|E_{Q_0}u|_2^2
+ =|p_0-e_{Q_0}|_2^2.
Diagonalize countably many tightness conditions; small
@@ -4799,61 +4799,61 @@ $ε_K$ gives the bound.
## Composition of admissible kernels
[Exact insertion]
-Let $\delta:L_\alpha(\Gamma)\to B_0\bar ⊗ L_\beta(\Gamma)$
+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
-$Θ\circledast\delta$ with source multiplier $\alpha$, whose
+$Θcircledastdelta$ with source multiplier $alpha$, whose
spherical Fourier laws are the exact compositions of the
-kernels of $\delta$ and $Θ$. The final measured cocycle
+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
+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
+$T_g(y)T_h(g^{-1}y)=nu(g,h)(y)T_{gh}(y)$ 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
+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_{h\in\Gamma}\|a_{\gamma,h}\|_2^2
- \int_Y\bigl\|D_{1_{\{r_T\in E\}}}Θ(c(h,h^{-1}y))\bigr\|_2^2\,dy
+ sum_{hinGamma}|a_{gamma,h}|_2^2
+ int_Ybigl|D_{1_{{r_Tin E}}}Θ(c(h,h^{-1}y))bigr|_2^2,dy
composes the kernels. Factoriality and admissibility come from
the induced coefficient representation
- (\mathcal U_gk)(y)=\Ad T_g(y)\,k(g^{-1}y),
+ (mathcal U_gk)(y)=calU_gk)(y)=Ad T_g(y),k(g^{-1}y),
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
- |< k',\mathcal U_i k>|,
+ sum_{i,j:,ih=h'j}|a_i|_2|a_j|_2
+ |< k',mathcal U_icalU_i k>|,
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 $\mathcal E$ by
-$L^2(Y;L^2(B(L^2(X_T))\bar ⊗ D))$ and $T_g$ by
-$\Delta_T(T_g)$, and admissibility of $Θ$ gives absence of
+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
invariant vectors.
## A linear subsequence
-Suppose $P_{Θ_n,R_n}(t_n)([a,∞])\ge\alpha$ with
-$a,\alpha>0$ and $u_n/t_n\to∞$. 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_∞,
+ P_{Θ_n,R_n}(u_n)→delta_∞,
uniformly under compact pre- and post-translations.
@@ -4863,26 +4863,26 @@ Proof.
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 $P([a,∞])\ge\alpha$ and
+limit $P^{*m}*Q$ with $P([a,∞])gealpha$ and
$Q$ supported in $[0,∞]$. For finite $M$,
- (P^{*m}*Q)([0,M])\le
- \mathbb P\{Bin(m,\alpha)\le M/a\}\to0
+ (P^{*m}*Q)([0,M])le
+ mathbb PbbP{Bin(m,alpha)le M/a}to0
-as $m\to∞$; 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 $C<∞$, $\alpha>0$, $T_n\to∞$ such that for
-every fixed $\xi\in\mathcal H$,
+exist $C<∞$, $alpha>0$, $T_nto∞$ such that for
+every fixed $xiinmathcal H$,calH$,
- liminf_{n}\inf_{T_n/2\le t\le T_n}
- \bigl\|D_{1_{\{r_{p(i)}\le CT_n\}}}U_{a_t}\xi\bigr\|^2
- \ge\alpha\|\xi\|^2.
+ liminf_{n}inf_{T_n/2le tle T_n}
+ bigl|D_{1_{{r_{p(i)}le CT_n}}}U_{a_t}xibigr|^2
+ gealpha|xi|^2.
The statement holds for normal Hilbert amplifications and
@@ -4891,14 +4891,14 @@ 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])\to0\qquad(C<∞).
+ P_{delta,t}(t)([0,C])to0qquad(C<∞).
Compose the original representation around the phase cycle
-$i\mapsto p(i)\mapsto...\mapsto i$ 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
@@ -4906,39 +4906,39 @@ 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
-$D'_n\to∞$: feasibility is nonempty (identity embedding
-into $L_\eta(\Gamma)$ suffices), so the infimum
+$D'_nto∞$: feasibility is nonempty (identity embedding
+into $L_eta(Gamma)$ suffices), so the infimum
- \inf\{t\ge1:\|D_{1_{\{r_i>L\}}}Θ(\gamma)\|_2\leε_K
- (\gamma\in\mathcal F),\ P_{Θ,D'_nL}(t)([1,∞])\ge\tfrac12\}
+ 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; $t_n\to∞$; and Lemma~lem:scale-many-increments
+$1/4$ mass; $t_nto∞$; and Lemma~lem:scale-many-increments
gives
- u_n/t_n\to∞⇒
- P_{Θ_n,R_n}(u_n)\to\delta_∞.
+ u_n/t_nto∞⇒
+ P_{Θ_n,R_n}(u_n)todelta_∞.
-Insert copies of $\delta$ for each cycle step. Choose
-$m_n\to∞$ with
+Insert copies of $delta$ for each cycle step. Choose
+$m_nto∞$ with
- t'_n=t_n/m_n\to∞,\quad
- P_{\delta,t_n}(t'_n)([m_n,∞])\to1,
+ t'_n=t_n/m_nto∞,quad
+ 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
coordinates stay tight; the terminal kernel is tested at length
-$\div t_n\to∞$, 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 $t_n\le2t'_n$.
+A tighter feasible witness $t'_n$ contradicts $t_nle2t'_n$.
Total superlinearity is therefore impossible. There are
-$C_0,\alpha_0>0$, $T_n\to∞$ with the scalar law mass
-$\ge\alpha_0$ below $C_0T_n$, uniform for $t\in[T_n/2,T_n]$
+$C_0,alpha_0>0$, $T_nto∞$ 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
@@ -4946,237 +4946,237 @@ 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 $\mathcal V_R$. The
+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(ab,x)=dil(a,bx)dil(b,x).
-All isolated factors here have $Q_T>2$. Let $\mathscr E\subset\mathcal H$
+All isolated factors here have $Q_T>2$. Let $mathscr EscrEsubsetmathcal H$calH$
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 $F\in C^∞(B_T\times B_T)$, its ordered evaluation is
-$\mathcal K_{x,y}(F)=\iint F(w,v)\,P_y(dw)P_x(dv)$ (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
-$d_zF|_{\mathcal V_T}$ vanishes on the diagonal.
+$d_zF|_{mathcal V_T}calV_T$ 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 $y=x\delta_ε w$ and $d\nu_ε=\chi(x)
-θ(w)\,dx\,dw$. For small $ε$:
-[label=(\roman*)]
-\item Order at least one gives, for each $\xi\in\mathscr E$,
-$\|\mathcal K_{x,y}(F)\xi\|_{L^2(d\nu_ε;\mathcal H)}
-\le C_{F,\xi}ε^{Q_S/Q_T}$.
-\item If $Q_S=Q_T=Q>2$ and order is at least two, the same norm is
+$chi$, 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
-$ε^{Q/2}$ ($2<Q<4$), $ε^2(1+|\logε|)$ ($Q=4$),
+$ε^{Q/2}$ ($2<Q<4$), $ε^2(1+|logε|)$ ($Q=4$),
$ε^2$ ($Q>4$).
Proof.
-The measure pullback under $w\mapsto y$ 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>\,d\nu_ε
- \le C_\xiε^{Q_S}\delta^{-Q_T},
+ 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 $c\delta\le d_T(w,v)\le C\delta$;
+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 $C\delta^m$. Expand in separated Fourier series, and
+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^m\min\{1,ε^{Q_S/2}\delta^{-Q_T/2}\},
+ C_{F,xi}delta^mmin{1,ε^{Q_S/2}delta^{-Q_T/2}},
-while the entire piece at scale $\delta_0$ has
-$C_F\delta_0^m\|\xi\|$. Summing over $\delta\geε^{Q_S/Q_T}$
+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 $\delta_0\asympε$ for the rates
+$Q_S=Q_T=Q$ 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 $f\in C^∞(B_T)$, and every
-$\xi\in\mathscr E$, the distributional derivative
-$D_X(f)\xi=X(x\mapsto P_x(f)\xi)$ lies in
-$L^2(B_S,dx;\mathcal H)$, linearly in $f,\xi$ and in smooth source
+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
+horizontal weight $psi_i$ with first moments the coordinate vector
gives difference quotients
-$ε^{-1}\int\psi_i(w)(P_{x\delta_ε w}(f)-P_x(f))\xi\,dw$
+$ε^{-1}intpsi_i(w)(P_{xdelta_ε w}(f)-P_x(f))xi,dw$
bounded in $L^2$ by $Cε^{Q_S/Q_T-1}$; adjoint pairing recovers
-the distributional derivative $-X\varphi-(divX)\varphi$.
+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
-$Q_S\le Q_T$; 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 $D_X(f)\xi\in L^2$ on charts; a partition of unity globalizes.
+give $D_X(f)xiin L^2$ on charts; a partition of unity globalizes.
-Set $\mathcal K=L^2(B_S,dx;\mathcal H)$ and
-$\mathcal C=span\{x\mapsto a(x)P_x(g)\xi:
-a\in L^∞(B_S), g\in C^∞(B_T), \xi\in\mathscr E\}$. For real
+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
+ 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 $\mathcal C$. 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)=0\quad\hbox{on }\mathcal C.
+ 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
-$(\Delta_{x,y}f)P_x(g)-P_y(g)\Delta_{x,y}f
-=\mathcal K_{x,y}((f(w)-f(v))(g(v)-g(w)))$; the right side has
+$(Delta_{x,y}f)P_x(g)-P_y(g)Delta_{x,y}f
+=mathcal K_calK_{x,y}((f(w)-f(v))(g(v)-g(w)))$; 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
+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>
+ < 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
pointwise form; complex weights follow by linearity. Substituting
-finitely many angular tests into a formal pairing of $u,v\in\mathcal C$
+finitely many angular tests into a formal pairing of $u,vinmathcal C$calC$
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 $\mathcal V_T$.)
+invertible on $mathcal V_T$.)calV_T$.)
There is a positive self-adjoint measurable field $l_x$ on
-$\mathcal H$, affiliated to $\mathcal A_1$, strongly commuting with
+$mathcal H$,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
- \beta_h(l_x)=dil(h^{-1},P_x)\,l_x,
+ sigma_a(l_x)=dil(a,x),l_{ax},qquad
+ beta_h(l_x)=dil(h^{-1},P_x),l_x,
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
-$\mathcal V_R$). Define the finite column
-$D:\mathcal C\to\mathcal K^{n_Sn_T}$,
-$Du=(T_{X_\alpha}(\theta_r)u)_{\alpha,r}$, closable by
-$\mathcal C$ density; set $l=|\overline D|$. Parameter and angular
-unitary multipliers commute with $D$ on $\mathcal C$, extending to
-the closed graph and form, so $l=\int^{ ⊕ }l_x\,dx$ with $l_x$
+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
complementary invariance.
-For covariance, note that Parseval systems embed $\mathcal V_S,\mathcal V_T$
+For covariance, note that Parseval systems embed $mathcal V_S,calV_S,mathcal V_T$calV_T$
isometrically into trivial bundles whose Gram matrices are orthogonal
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 $\mathcal C$;
+(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
+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 $\mathcal A_1$, strongly commuting
+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
- \beta_h(b_x)=dil(h^{-1},P_x)^{-Q}b_x.
+ sigma_a(b_x)=dil(a,x)^{-Q}b_{ax},qquad
+ 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 $x_0\in B_S$ with parabolic
-stabilizer $J$, pull back along $g\mapsto gx_0$ and set
-$F(g)=dil(g,x_0)\sigma_g^{-1}(l_{gx_0})$. Haar
+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 $a\in S$; Fubini over
-$(a,g)\mapsto(ag,g)$ reduces $F$ to an essential value $L_0$ (test
+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
-$k\in J$, $F(gk)=dil(k,x_0)\sigma_k^{-1}F(g)$, 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
-$e_0\in\mathcal N$. Source stabilizer covariance gives
-$\sigma_k(e_0)=e_0$. A split sequence in $J$ escaping in $S$ times
+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 $e_0\in\{0,1\}$. Nonzero column forces $e_0=1$ (this
+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
uses complementary source invariance essentially).
Every transported $l_x$ has trivial kernel; define $b_x=l_x^{-Q}$ by
spectral calculus. Dense, positive, self-adjoint, injective, affiliated
-to $\mathcal A_1$; 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 $\mathcal H ⊗ \mathcal L$ with
-orthonormal basis $(e_n)$, an element $\Xi=\sum_n\xi_n ⊗ e_n$ of
+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
the maximal amplified estimate domain has every coordinate
-$\xi_n\in\mathscr E$ (each coordinate energy is bounded by the
-total); conversely finite sums of $\xi_n\in\mathscr E$ lie in the
+$xi_ninmathscr E$scrE$ (each coordinate energy is bounded by the
+total); conversely finite sums of $xi_ninmathscr E$scrE$ 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 $\mathcal C\odotspan\{e_n\}$ 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
+$l^{rm amp}=l ⊗ 1$ and $b^{rm amp}=b ⊗ 1$; unitary
identification with a normal copy preserves the identities, without
any mixing on an enlarged commutant.
@@ -5216,14 +5216,14 @@ 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
+completing simple gap at the selected panel and $ell(g)$ the residual
target Cartan gap, then
- d_T\ell(g)\le d_St+O_{tight}(1).
+ d_Tell(g)le d_St+O_{tight}(1).
Here tight upper error means the event where
-$d_T\ell(g)-d_St$ 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
(eq:input-11) is the resulting shifted gap. Gap signs are the
@@ -5266,11 +5266,11 @@ 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
-\[
- V_t=\ker(d\pi(H_R)-t),\qquad
- h(V)=\sum_{t>0}\dim V_t,\qquad z(V)=\dim V_0.
-\]
-For a nontrivial irreducible $V$, $\dim V/h(V)$ is the inverse
+[
+ V_t=ker(dpi(H_R)-t),qquad
+ h(V)=sum_{t>0}dim V_t,qquad z(V)=dim V_0.
+]
+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
@@ -5288,46 +5288,46 @@ reduced restricted root in both cases.
[Optimal panel modules]
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 $c_R=\max_V\dim V/h(V)$ and all
+Lie-algebra modules, the maximum $c_R=max_Vdim V/h(V)$ and all
modules attaining it are
-\[
+[
{ccl}
- R&c_R&maximizing modules\\ \hline
- SO(n,1),\ n\ge2 & k & E\\
- SU(n,1),\ n>2 & k & E,E^*\\
- SU(2,1) & 3 & E,E^*,Sym^2E,Sym^2E^*\\
- Sp(n,1),\ n>2 & k & E\\
- Sp(2,1) & 7/2 & ∧\nolimits^3_0E
-
-\]
+ R&c_R&maximizing modules hline
+ SO(n,1), nge2 & k & E
+ SU(n,1), n>2 & k & E,E^*
+ SU(2,1) & 3 & E,E^*,Sym^2E,Sym^2E^*
+ Sp(n,1), n>2 & k & E
+ Sp(2,1) & 7/2 & ∧nolimits^3_0E
+
+]
where $∧^3_0E$ is the primitive exterior cube (kernel of the
-symplectic contraction $∧^3E\to E$). There is
+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
+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 $d\pi(H_R)^*=d\pi(H_R)$. The form pairs
+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
-$V_{\ge0}=⊕_{t\ge0}V_t$,
-\[
- V_{>0}^{\perp_q}=V_{\ge0},\qquad
- V_{\ge0}^{\perp_q}=V_{>0},
-\]
-and the induced form on $V_{\ge0}/V_{>0}$ is positive definite. These
+$V_{ge0}=⊕_{tge0}V_t$,
+[
+ V_{>0}^{perp_q}=V_{ge0},qquad
+ V_{ge0}^{perp_q}=V_{>0},
+]
+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
-\[
- d_R=2,&R=SU(2,1) or Sp(2,1),\\1,&otherwise,
-\]
+[
+ d_R=2,&R=SU(2,1) or Sp(2,1),1,&otherwise,
+]
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
+$n_0=dim V$, 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
@@ -5336,11 +5336,11 @@ conjugates; every summand has the same optimal ratio, so still
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
+compact projectivity frames choose local or Borel lifts $pi_T(g)$ of
target projectivities such that
- \pi_T(g)^*Q\pi_T(g)=Q,\qquad
- \|\pi_T(g)\|=\exp(d_T\ell(g)).
+ pi_T(g)^*Qpi_T(g)=Q,qquad
+ |pi_T(g)|=exp(d_Tell(g)).
The connected action lifts to the simply connected module; compact