diff 56bc728..f6bf760 arithmeticity_local_field_lattices_v2.txt
diff --git a/arithmeticity_local_field_lattices_v2.txt b/arithmeticity_local_field_lattices_v2.txt
index ec76ea8..a3f71b0 100644
--- a/arithmeticity_local_field_lattices_v2.txt
+++ b/arithmeticity_local_field_lattices_v2.txt
@@ -94,8 +94,8 @@ Some of the most important groups are continuous: instead of
finitely many or countably many symmetries, they form a smooth
space. For example, the set of all rotations of the plane about the
origin is a circle; the set of all invertible ntimes n real matrices
-of determinant one, written SL_n(mathbb R),bbR), is a continuous group of
-transformations of mathbb R^nbbR^n that preserve volume and orientation.
+of determinant one, written SL_n(R), is a continuous group of
+transformations of R^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.
@@ -103,11 +103,11 @@ 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^nbbZ^nsubsetmathbb R^nbbR^n
+ Z^n R^nbbR^n
]
inside the group of all translations, or the famous
[
- SL_n(mathbb Z)bbZ)subsetSL_n(mathbb R)bbR)
+ SL_n(Z) R)bbR)
]
of integer matrices of determinant one inside all real matrices of
determinant one. A lattice is discrete, so its elements do not
@@ -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,bbR, the complex numbers mathbb C,bbC,
-and the p-adic numbers mathbb Q_pbbQ_p are all "completions" of the
+theory: the real numbers R, the complex numbers C,
+and the p-adic numbers Q_p are all "completions" of the
rationals, and a formula that holds over the rationals has a face at
each place. A "local field" here means a finite extension of
-mathbb R,bbR, mathbb C,bbC, or mathbb Q_p.bbQ_p. Real place, complex place,
+R, C, or Q_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,scrK, and it consists of the countably infinite groups that are
+K, 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
points of a connected, adjoint, absolutely simple algebraic group over a
-local field k_iin{mathbb R,bbR,mathbb C,finitebbC,finite extension of mathbb Q_pbbQ_p},
+local field k_iin{R,C,finitebbC,finite extension of Q_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,
@@ -178,7 +178,7 @@ 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)~=
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.
@@ -197,11 +197,11 @@ Sometimes the symmetries of a system are only defined up to a phase: a
rotation by the identity angle might act as multiplication by a complex
number of modulus one. Such an "almost action" is a projective
representation, and its failure of multiplicativity is encoded in a
-cocycle mu:GammatimesGammatomathbb T.bbT. The rule is
+cocycle mu:GammatimesGammatoT. 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
+with values in the unit circle 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}.
Scalar cocycles are unavoidable in the geometry of these problems, and a
@@ -248,7 +248,7 @@ KscrK and arbitrary countably infinite ICC Λ, is it built out of
arithmetic models? The answer proved here is yes.
-Main theorem (Arithmetic exhaustion). Let Gammainmathscr K,scrK,
+Main theorem (Arithmetic exhaustion). Let GammainK,
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
@@ -435,13 +435,13 @@ group is available on the comparison side at the start.
## The class, conventions, and main theorem
-Let mathscr KscrK be the class of countably infinite ICC groups abstractly
+Let K 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, 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_ibfH_i is
+Here k_i is R, C, or a finite extension of
+Q_p; the primes may differ. The algebraic group mathbf H_ibfH_i is
connected, adjoint, and absolutely simple over k_i. The superscript +
denotes the subgroup generated by the k_i-points of the unipotent
radicals of proper k_i-parabolic subgroups. Each G_i is required to be
@@ -463,7 +463,7 @@ differential arguments are used. Standard structure and permanence
results are recalled in Section~sec:setup; see also
[cite].
-All scalar cocycles are mathbb T-valuedbbT-valued and normalized:
+All scalar cocycles are T-valuedbbT-valued and normalized:
[
mu(g,h)mu(gh,j)=mu(h,j)mu(g,hj),
mu(e,g)=mu(g,e)=1.
@@ -482,7 +482,7 @@ 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)bbC)barbar ⊗⊗ Q)p,
+ Q^t=p(M_n(C) ⊗⊗ Q)p,
(Tr_n ⊗ tau_Q)(p)=t,
]
with the unnormalized matrix trace Tr_n; the corner itself carries its
@@ -514,9 +514,9 @@ U_{(g,h)}xi=u_gxi v_h^* has multiplier
alpha((g,h),(g',h'))=mu(g,g')omega(h,h')bar.
[Arithmetic exhaustion]
-Let Gammainmathscr K,scrK, let Λ be any countably infinite ICC
-group, and let muin Z^2(Gamma,mathbb T),bbT),
-omegain Z^2(Λ,mathbb T)bbT) be normalized scalar cocycles.
+Let GammainK, let Λ be any countably infinite ICC
+group, and let muin Z^2(Gamma,T),
+omegain Z^2(Λ,T) 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),
@@ -527,7 +527,7 @@ Equation~(eq:intro-multiplier).
Conversely, every such model, finite direct sum, and closed bimodule
summand is a normal bifinite correspondence. If
-n=dim_{mathbb C}bbCV_sigma, the elementary model has dimensions
+n=dim_{C}, the elementary model has dimensions
[
dim_{L_mu(Gamma)}E(A,B,delta,sigma)=n[Λ:B],
dim E(A,B,delta,sigma)_{L_omega(Λ)}=n[Gamma:A].
@@ -605,7 +605,7 @@ local-field lattice class.
A geometric line of operator-algebraic rigidity for lattices begins with
character and finite-factor representation results. Bekka classified the
-characters of SL_n(mathbb Z),bbZ), nge3 cite[Theorem~3]{Bekka2007};
+characters of SL_n(Z), 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
@@ -663,7 +663,7 @@ 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.
+ for 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
@@ -780,7 +780,7 @@ projective representation on a finite-dimensional space V, with
multiplier alpha|_{Ctimes C}. Write X=D/C for the left cosets,
choose representatives r_xin D with r_C=e, and give each xin X a
copy V_x of V. Set
-mathcal H(C,calH(C,gamma)=⊕_{xin X}V_x, with the counting Hilbert
+H(C,gamma)=⊕_{xin X}V_x, with the counting Hilbert
norm. For sin D put
@@ -813,7 +813,7 @@ of gamma in Equation~(eq:input-A1) gives
Equivalently, all scalar factors can be verified at once in the central
extension with multiplication (z,s)(w,t)=(zwalpha(s,t),st): the
-representation of the preimage of C is (z,c)mapsto zgamma(c), and
+representation of the preimage of C is (z,c)|-> 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.
@@ -840,13 +840,13 @@ 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
+representations of the right generators xi|->xi v_h^*. The two
actions commute, since the product cocycle has trivial mixed-coordinate
factors.
The first-coordinate orbits are indexed by Λ/B and the
second-coordinate orbits by Gamma/A; both are finite. Each orbit is a
-regular module with multiplicity dim_{mathbb C}bbCV, proving bifiniteness
+regular module with multiplicity dim_{C}, proving bifiniteness
and the dimension formulas below.
@@ -858,14 +858,14 @@ the model.
[The graph-bundle unitary]
Let delta:Ato B be a finite-index subgroup isomorphism and let
-sigma:Atomathcal U(V)calU(V) have multiplier
+sigma:AtoU(V) have multiplier
beta=mu|_{Atimes A}/delta^*omega. Set
gamma(a,delta(a))=sigma(a). Then
[
E(A,B,delta,sigma)=
L^2(M)mathbin{barbar ⊗⊗ }_{L_{mu|A}(A)}
(V ⊗ L^2(N))
- congmathcal H(C_calH(C_delta,gamma).
+ congH(C_,gamma).
]
The unitary sends the vector xi in the fibre labelled x to
U_{r_x}jxi, where
@@ -936,7 +936,7 @@ Equation~(eq:input-A1).
[Elementary normality and dimensions]
The correspondence E(A,B,delta,sigma) is normal and bifinite. If
-n=dim_{mathbb C}bbCV_sigma, then
+n=dim_{C}, then
[
dim_M E(A,B,delta,sigma)=n[Λ:B],
dim E(A,B,delta,sigma)_N=n[Gamma:A].
@@ -992,7 +992,7 @@ lattice Gamma<G and under the condition that
Adrho on
L^2(pi(L_mu(Gamma))',tr)
- ominusmathbb C1bbC1
+ ominusC1
quadhas no nonzero finite-dimensional subrepresentation.
@@ -1007,7 +1007,7 @@ Let alpha,nu be normalized scalar cocycles on a countable group L,
and let v be a nu-projective unitary representation on a Hilbert
space V. Then
[
- lambda_alpha ⊗ vsimeqlambda_{alphanu} ⊗ 1_V.
+ lambda_alpha ⊗ v~=lambda_{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
@@ -1079,7 +1079,7 @@ 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)ominusmathbb C1bbC1 has no nonzero
+Twisted conjugation on L^2(N)ominusC1 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
@@ -1095,11 +1095,11 @@ projections, and is finite-dimensional.
Restrict to the finite-index subgroup Λ_0 fixing all central
blocks of D_c. A central block z reduces both endpoint actions; write
-zK=V_c ⊗ K'_c and zD_c=mathcal B(V_c)calB(V_c) ⊗ 1. The left action
+zK=V_c ⊗ K'_c and zD_c=B(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
+implementers rho(h) lies in the commutant of B(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
@@ -1168,7 +1168,7 @@ class in Gamma is a finite union of translates of that class; ICC of
Gamma forces it to be the identity.
-For a source in mathscr K,scrK, choose the actual isomorphism of finite-index
+For a source in K, 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
@@ -1229,12 +1229,12 @@ slot.
Write P=pi(M)'. The final joint commutant is
(P^{barbar ⊗⊗ (m+1)})^{Adrho^{ ⊗ (m+1)}}, which
Equation~(eq:setup-weak-mixing) makes scalar. For completeness: an
-invariant tensor with a factor in L^2(P)ominusmathbb C1bbC1 gives a
+invariant tensor with a factor in L^2(P)ominusC1 gives a
Hilbert--Schmidt intertwiner between that factor's representation and
another unitary representation; composing with its adjoint gives a
positive compact operator with a nonzero finite-dimensional invariant
spectral space, contrary to weak mixing. Decomposing each tensor factor
-as mathbb C1bbC1 ⊕ (L^2(P)ominusmathbb C1)bbC1) applies this to every
+as C1 ⊕ (L^2(P)ominusC1) applies this to every
nonconstant summand.
@@ -1250,12 +1250,12 @@ w_r. Thus W_r is right multiplication by w_r^*, which has
multiplier alpha_R. Set
Y=G/Gamma,
- mathcal H=L^2(Y;H),calH=L^2(Y;H),
- mathcal N=L^∞(Y)calN=L^∞(Y)overlineoverline ⊗⊗ eM_k(B)e.
+ H=L^2(Y;H),calH=L^2(Y;H),
+ N=L^∞(Y)calN=L^∞(Y)overlineoverline ⊗⊗ eM_k(B)e.
-Give Y its invariant probability measure. The trace on mathcal NcalN is
-tr_{mathcal N}calN(A)
+Give Y its invariant probability measure. The trace on N is
+tr_{N}
=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
@@ -1278,15 +1278,15 @@ sigma=Ad U.
[Induced finite algebras]
The transports U_a are strongly continuous in a, and sigma
-preserves the trace on mathcal N.calN. For 1le jle m, let R'_j be R
+preserves the trace on N. For 1le jle m, let R'_j be R
with Gamma_j omitted, put beta_j=Ad W_{Gamma_j}, and let
-mathcal A_jcalA_j be the commutant of the base multipliers and
+A_j be the commutant of the base multipliers and
W|_{R'_j}. Then
[
- (mathcal A_j)^calA_j)^{beta_j}=mathcal N,calN,
- (mathcal A_j)^calA_j)^sigma has a faithful finite beta_j-invariant
+ (A_j)^{beta_j}=N,
+ (A_j)^ has a faithful finite beta_j-invariant
trace,
- mathcal N^G=calN^G=mathbb C1.bbC1.
+ N^G= C1.bbC1.
]
@@ -1297,7 +1297,7 @@ c(a,y) varies continuously in measure as a discrete-valued map. Verify
first on compact sets where the section is continuous, by Lusin
approximation, then remove sets of small measure. The corresponding
projective unitaries are bounded, so their multiplication operators vary
-strongly on mathcal H.calH. Conjugation of the corner is inner at each fibre
+strongly on H. 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.
@@ -1307,12 +1307,12 @@ for g=d_0(y)gamma, as widetilde A(g)=pi_0(gamma)^*A(y)pi_0(gamma).
The cocycle covariance makes this field left G-invariant, hence almost
everywhere constant with value T, and right Gamma-covariance says T
commutes with pi_0(Gamma) and with W|_{R'_j}. Conversely every such
-T gives a fixed field. Thus (mathcal A_j)^calA_j)^sigma is the commutant of
+T gives a fixed field. Thus (A_j)^ 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
that commutant. Including all slots and using the last assertion of that
-lemma gives mathcal N^G=calN^G=mathbb C1.bbC1.
+lemma gives N^G= C1.bbC1.
## The source groups and full versus partial mixing
@@ -1359,11 +1359,11 @@ This proves the finite-index assertion directly.
[Product mixing and partial escape]
-For the action on L^2(mathcal N),calN), escape in every simple factor gives
+For the action on L^2(N), escape in every simple factor gives
weak operator convergence to the projection onto constants. If a path is
the identity outside a subproduct G_0 and escapes in every factor of
-G_0, its weak limit is the projection onto L^2(mathcal N)^calN)^{G_0},
-equivalently the conditional expectation onto mathcal N^calN^{G_0}. On the
+G_0, its weak limit is the projection onto L^2(N)^{G_0},
+equivalently the conditional expectation onto N^{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.
@@ -1380,7 +1380,7 @@ radical mathbf U.bfU. The k-span of the mathbf H(k)-conjugatesbfH(k)-conjugates
Lie(mathbf U)bfU) is a nonzero invariant Lie ideal, hence all
of Lie(mathbf H).bfH). Choose X_1,...,X_d from these
conjugate subspaces; the map
-(t_1,...,t_d)mapstoprod_jexp(t_jX_j) has invertible differential
+(t_1,...,t_d)|->prod_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
@@ -1395,7 +1395,7 @@ cite[Proposition~6.E.21(1)]{BekkaHarpe2019}.
Restrict a separable unitary representation pi to one simple factor S.
Its type-I decomposition with multiplicity is
[
- mathcal H=calH=int_{widehatwidehat S}S^{ ⊕ }(H_xi ⊗ M_xi),deta(xi),
+ H={widehatwidehat S}S^{ ⊕ }(H_xi ⊗ M_xi),deta(xi),
pi(s)=int_{widehatwidehat S}S^{ ⊕ }(xi(s) ⊗ 1),deta(xi).
]
@@ -1422,7 +1422,7 @@ decay kills its coefficients when that factor escapes. Thus all-factor
escape gives the weak limit prod_iP_i, the projection onto constants
by Proposition~prop:setup-induced-algebras. For partial escape,
expand only over the factors of G_0; the limit is the projection onto
-L^2(mathcal N)^calN)^{G_0}, i.e. the trace-preserving conditional
+L^2(N)^{G_0}, i.e. the trace-preserving conditional
expectation.
@@ -1436,9 +1436,9 @@ An archimedean panel inside a higher-rank component cannot have noncompact
Lie algebra mathfrak f_frakf_{4(-20)}: its complexification would have to
occur as a component of a proper absolute Dynkin subdiagram, and the
Dynkin list has no such occurrence. At a finite place the effective
-rank-one Lie algebra is simple also over mathbb Q_p:bbQ_p: after restriction
+rank-one Lie algebra is simple also over Q_p:bbQ_p: after restriction
of scalars its absolutely simple factors form one Galois orbit, so a
-mathbb Q_p-idealbbQ_p-ideal is either zero or the whole algebra. An adjoint
+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
@@ -1632,7 +1632,7 @@ 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
+group. A target Fourier arrow records w_{(h,·)} 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
@@ -1651,14 +1651,14 @@ translation produces exactly one divergent target gap while successively
completing such a vertex-origin path adds exactly one target gap per stage.
We retain the regular realizations of Section~sec:setup: H=eL^2(B)^k,
-mathcal H=L^2(Y;H),calH=L^2(Y;H), d=(Tr ⊗ tau_B)(e). For an integrable corner
+H=L^2(Y;H),calH=L^2(Y;H), d=(Tr ⊗ tau_B)(e). For an integrable corner
field write
[
- mathsf T(C)=sfT(C)=int_Y(Tr ⊗ tau_B)(C(y)),dy,
- tau_{mathcal N}calN(C)=d^{-1}mathsf T(C).sfT(C).
+ T(C)= ⊗ tau_B)(C(y)),dy,
+ tau_{N}{-1}T(C).
]
-Thus tau_{mathcal N}calN is a probability trace while the Hilbert norm of a
-column is |f|_2^2=mathsf T(ff^*).sfT(ff^*). Matrix sizes in this section are
+Thus tau_{N} is a probability trace while the Hilbert norm of a
+column is |f|_2^2=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
[
@@ -1767,7 +1767,7 @@ The one-column map
(V_a f)(y)=X_a(y)f(y)
]
is an isometry with the integrated tracial Hilbert norm, so its Hilbert
-ultralimit V is an isometry. On the minimal spectral span of Vmathcal HcalH
+ultralimit V is an isometry. On the minimal spectral span of VH
the continuous diagonal tests give a PVM E^{(1)}; the compression
[
Phi(A)=V^*E^{(1)}(A)V
@@ -1861,29 +1861,29 @@ proposition, not an assumption on the construction.
The one-column measurements and the forms (eq:input-1) have the
following properties.
[label=(roman*)]
- The image measurement in one measured copy lies in mathcal A_j,calA_j,
+ The image measurement in one measured copy lies in A_j,
is beta_j-equivariant, and is invariant under the source face parabolic.
-Its gap-stratum masses belong to mathcal NcalN and are scalar. Conjugating
+Its gap-stratum masses belong to N and are scalar. Conjugating
the source split frame transports the measurement by sigma.
If the path escapes in every simple component, the link is an isometry
-mathcal HbarcalHbar ⊗ mathcal HcalHtomathcal K.calK. Its output forms are
+Hbar ⊗ H 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
rule is symmetric.
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
+N_0=N^{G_0} and let E_{N_0}:N N_0 be the
trace-preserving conditional expectation. On bounded columns the link has
|J(overlineoverline b,f)b,f)|^2
- =d^2tau_{mathcal N}calN(E_{N_0}(ff^*)E_{N_0}(bb^*)).
+ =d^2tau_{N}{N_0}(ff^*)E_{N_0}(bb^*)).
Its polarized extension on finite sums,
[
Bigl|sum_iJ(b_ibar,f_i)Bigr|^2
- =d^2sum_{i,l}tau_{mathcal N}calN
+ =d^2sum_{i,l}tau_{N}
(E_{N_0}(f_lf_i^*)E_{N_0}(b_ib_l^*)),
]
@@ -1915,7 +1915,7 @@ translate must tend to the identity in that order.
Proof.
(i) Base multipliers and companion actions commute with the relevant
diagonal tests and intertwine the input column map, so the compressed image
-effects lie in mathcal A_j.calA_j. A measured target shift changes the inverse
+effects lie in A_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.
@@ -1934,30 +1934,30 @@ the uncut Gram factor is the corner identity. Thus the image measurement is
centralizer invariant. Radical and centralizer together generate the face
parabolic; conjugation of a split path removes the end conjugation.
-The mass of a gap stratum is target invariant, hence in mathcal N,calN, and is
+The mass of a gap stratum is target invariant, hence in N, and is
invariant under the entire face parabolic. Every inactive simple factor is
in the centralizer; in an active factor the contracted radical contains an
escaping sequence. By Howe--Moore applied to the representation on
-L^2(mathcal N),calN), a radical-invariant vector orthogonal to the
+L^2(N), a radical-invariant vector orthogonal to the
factor-invariant subspace would have a matrix coefficient both constant and
tending to zero. It is therefore factor invariant, and the mass is invariant
-under all simple factors, hence scalar because mathcal N^G=calN^G=mathbb C1.bbC1.
+under all simple factors, hence scalar because N^G= 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(overlineoverline b,f)b,f)|^2
- =dlim_amathsf T(ff^*sfT(ff^*sigma_{a^{-1}}(bb^*)).
+ =dlim_aT(ff^*sfT(ff^*sigma_{a^{-1}}(bb^*)).
]
On a mixed path, trace mixing makes the limit
-d^2tau_{mathcal N}calN(ff^*)tau_{mathcal N}calN(bb^*)=|f|_2^2|b|_2^2.
+d^2tau_{N}*)tau_{N}*)=|f|_2^2|b|_2^2.
Polarization gives
[
Bigl|sum_iJ(b_ibar,f_i)Bigr|^2
- =d^2sum_{i,l}tau_{mathcal N}calN(f_lf_i^*)tau_{mathcal N}calN(b_ib_l^*),
+ =d^2sum_{i,l}tau_{N}*)tau_{N}*),
]
-which is the norm on mathcal HbarcalHbar ⊗ mathcal H;calH; the factor
+which is the norm on Hbar ⊗ 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
@@ -1967,12 +1967,12 @@ bases. Independent target shifts are handled by Fourier transport; source
radicals slide to contracted translates and cancel; simultaneous
centralizers use exact precomposition and finite-Fourier Gram. For an
image-and-stratum event, the ignored end Fourier shifts put each end slice
-in mathcal Nbar,calNbar, and independent reverse-radical invariance makes
+in Nbar, 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).
(iii) For an escape in G_0, the weak limit of the trace-preserving
-action on L^2(mathcal N)calN) is the projection onto L^2(N_0), which yields
+action on L^2(N) 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.
@@ -2067,7 +2067,7 @@ algebra, its support projections, and after finite matrix amplification.
For a closed subspace invariant under N_0, the spectral projections
p_n=1_{[0,n]}(E_{N_0}(ff^*)) preserve it; p_nf is relatively bounded and
converges to f since
-|(1-p_n)f|_2^2=dtau_{mathcal N}calN((1-p_n)E_{N_0}(ff^*))to0. For
+|(1-p_n)f|_2^2=dtau_{N}-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^*).
@@ -2107,12 +2107,12 @@ A fixed Fourier shift gives precisely the limiting intertwining identity
used in Lemma~lem:links-gram-transport; the finite-Fourier proof is
unchanged. Base decomposability and commutation with companions follow
before limits. An invariant one-column test gives an element of
-mathcal N;calN; face-parabolic invariance, followed by the active-factor
+N; face-parabolic invariance, followed by the active-factor
radical argument and the inactive-factor centralizer argument of
-Proposition~prop:links-link, puts it in mathcal N^G=calN^G=mathbb C1.bbC1.
+Proposition~prop:links-link, puts it in N^G= C1.bbC1.
For mixed links, independent target invariance puts the respective slices
-in mathcal NbarcalNbar and mathcal N;calN; independent
+in Nbar and N; independent
contracted-radical invariance makes them scalar, and polarization and
normal slice separation give the scalar joint effect.
@@ -2179,7 +2179,7 @@ the two inputs. The generator identification of minimal spectral dilations
gives the enlarged dilation. A face spectral measure invariant under the
projective transport now commutes with a genuine source representation and
can be disintegrated with it. All mixing and scalarity statements continue
-to refer to mathcal N,calN, not the enlarged commutant.
+to refer to N, not the enlarged commutant.
## Escape from bounded target coordinates
@@ -2245,23 +2245,23 @@ 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_jcalA_j which is target-equivariant and fixed by the
+space Z to A_j which is target-equivariant and fixed by the
source action sigma|_S.
Proof.
-Suppose Phi:C(Z)tomathcal A_jcalA_j is such a map. Its mass a=Phi(1)
-belongs to mathcal N^S:calN^S: target equivariance makes it commute with the
-measured target action, and membership in mathcal A_jcalA_j the rest. Choose
-ε>0 with p=1_{[ε,∞)}(a)ne0. On pmathcal H,calH,
+Suppose Phi:C(Z)toA_j is such a map. Its mass a=Phi(1)
+belongs to N^S:calN^S: target equivariance makes it commute with the
+measured target action, and membership in A_j the rest. Choose
+ε>0 with p=1_{[ε,∞)}(a)ne0. On pH,
Psi(u)=pa^{-1/2}Phi(u)a^{-1/2}p is unital and completely positive (a
positive map from C(Z) is completely positive) with the same covariances.
The normalization does not require a to commute with Phi(u).
-Choose a discrete free Fcongmathbb F_2bbF_2subset S, obtained from two
+Choose a discrete free FcongF_2 S, obtained from two
sufficiently strong rank-one contractions with disjoint attracting and
-repelling neighborhoods. Let mathcal DcalD be the concrete C^*-algebra on
-pmathcal HcalH generated by cut base multipliers, the companion actions
+repelling neighborhoods. Let D be the concrete C^*-algebra on
+pH 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.
@@ -2272,7 +2272,7 @@ vectors u ⊗ xi with form
=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
+T_g(u ⊗ xi)=(g· u) ⊗ W_gxi has the same multiplier
zeta as W.
By cite[Proposition~2.5]{AnantharamanDelaroche2002}, the action of
@@ -2282,7 +2282,7 @@ to 1, and uniform asymptotic equivariance. Compactness of Z makes their
norms uniformly positive eventually, and normalizing pointwise gives
continuous unit fields with the same uniformity. In the dilation with
spectral representation pi_Z of C(Z), put
-Q_ieta=sum_{tinGamma_j}delta_t ⊗ pi_Z(xi_i(cdot)_t)eta;
+Q_ieta=sum_{tinGamma_j}delta_t ⊗ pi_Z(xi_i(·)_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
@@ -2292,7 +2292,7 @@ level,
Bigl|sum_g W_gD_gBigr|
leBigl|sum_glambda_zeta(g) ⊗ D_gBigr|_{
- C^*_{r,zeta}(Gamma_j) ⊗ _{min}mathcal D}calD.
+ C^*_{r,zeta}(Gamma_j) ⊗ _{min}D}
The multiplier is that of W and is conjugated in the right-action
@@ -2300,7 +2300,7 @@ convention.
Complementary regularity makes H a subrepresentation of a finite regular
multiple for Gamma_0times R'_j with split multiplier. Inducing the
-Gamma_0-coordinate and using (y,h)mapsto d_0(y)h turns it into the
+Gamma_0-coordinate and using (y,h)|-> 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),
@@ -2317,7 +2317,7 @@ We contradict this on one matrix vector. Regard p as k columns and put
T_f(y)=p(x)X_f(x)^*p(y) with x=f^{-1}y; this is a contraction. Since
p is sigma_f-invariant, X_f(x)p(x)X_f(x)^*=p(y), and right
multiplication after the shift gives
-(R_{T_f}U_f^{(k)}p)(y)=X_f(x)p(x)cdot p(x)X_f(x)^*p(y)=p(y). For the
+(R_{T_f}U_f^{(k)}p)(y)=X_f(x)p(x)· 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
@@ -2327,7 +2327,7 @@ Then S_n=sum_fnu(f)R_{T_{f,n}}U_f^{(k)} has S_nEto Ene0. By
|S_n|le|lambda_F(nu)|; for the uniform walk on two free generators
and their inverses this is at most sqrt3/2<1 by Kesten's bound
cite[Theorem~3]{Kesten1959}. The conclusions are incompatible since
-0<|E|_2^2=mathsf T(p)<∞.sfT(p)<∞.
+0<|E|_2^2=T(p)<∞.sfT(p)<∞.
## Strict growth under a slower Levi translation
@@ -2352,22 +2352,22 @@ Lemma~lem:links-obstruction. Thus DnePi.
The marginal inclusion. Let Psi(F)=V_Ω^*M_FV_Ω be a
minimal full-data dilation with image-flag map x:Ω_Dto X_D and
-minimal image dilation V:mathcal HcalHtomathcal K_X.calK_X. The generator map
-M_uVfmapsto M_{ucirc x}V_Ω f preserves inner products and extends
-to an isometric inclusion i:mathcal K_XcalK_Xtomathcal K_ΩcalK_Ω with
+minimal image dilation V:H K_X.calK_X. The generator map
+M_uVf|-> M_{ucirc x}V_Ω f preserves inner products and extends
+to an isometric inclusion i:K_X K_ΩcalK_Ω with
iV=V_Ω, intertwining the image spectral measure, base multiplication,
companions, and target lifts. Disintegrate
-mathcal K_X=calK_X=int^ ⊕ K_x,dθ(x) with isometric fibre
+K_X= ⊕ K_x,dθ(x) with isometric fibre
inclusions i_x.
-Positive density forms. Let mathcal F_xcalF_x be the compact
+Positive density forms. Let F_x be the compact
full-refinement space of the image face x, with compact-rotation
probability eta_x. The residual projectivity of a full datum
omegainΩ_D transports that probability to a probability
-k_omega(l),deta_x(l) on mathcal F_x,calF_x, positive with total mass one.
+k_omega(l),deta_x(l) on F_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
+linF_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
densities in finitely many compact-open charts.
@@ -2377,28 +2377,28 @@ D_{n,x}(l)=i_x^*M_{min(k_omega(l),n)}i_x. The limiting quadratic form
q_{x,l}(v)=|M_{√(k_omega(l))}i_xv|^2 is closed. Its domain is dense
for almost every (x,l): choose f_r with Vf_r(x) total in K_x
almost everywhere (the measurable projection onto the fibrewise orthogonal
-complement kills angular multiples of Vmathcal H,calH, so vanishes by
+complement kills angular multiples of VH, 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)
+ int_{X_D}int_{F_x}{x,l}(Vf_r(x)),deta_x(l)dθ(x)
=|f_r|^2.
]
For the associated positive D_x(l), measurability follows from the
-bounded approximants, and int_{mathcal F_x}calF_xq_{x,l}(v),deta_x(l)=|v|^2
+bounded approximants, and int_{F_x}{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 HcalH to
-mathcal R=calR=int^ ⊕ _{X_D}L^2(mathcal F_x,calF_x,eta_x;K_x)dθ(x).
+is a bounded isometry from H to
+R= ⊕ _{X_D}L^2(F_x,eta_x;K_x)dθ(x).
Equivariance. For a target gamma, set
r_{gamma,x}(l')=d(gamma_*eta_x)/deta_{gamma x}(l'). Then
k_{gammaomega}(gamma l)=r_{gamma,x}(gamma l)k_omega(l), and since
i intertwines target lifts, the compressed form is
D_{gamma x}(gamma l)=r_{gamma,x}(gamma l)u_{gamma,x}D_x(l)u_{gamma,x}^*
-for the fibre-unitary u_{gamma,x}; the target lift on mathcal RcalR
+for the fibre-unitary u_{gamma,x}; the target lift on R
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
@@ -2406,34 +2406,34 @@ 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 RcalR by
+U_s^X(M_uVf)=M_uVU_sf; extend it to widetilde U_s on R 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_0scrE_0 be the bounded
-intertwiners mathcal HcalHtomathcal RcalR for the base and target actions;
-B^*Cinmathcal NcalN for B,Cinmathscr E_0,scrE_0, and
-< B,C>_{mathscr E}scrE=tau_{mathcal N}calN(B^*C) completes to
-mathscr EscrE with
+The finite-trace intertwiner space. Let E_0 be the bounded
+intertwiners H RcalR for the base and target actions;
+B^*CinN for B,CinE_0, and
+< B,C>_{E}{N}*C) completes to
+E with
- |B|_{mathscr E}scrE^2=frac1dsum_{r=1}^k|Bb_r|^2,
+ |B|_{E}{r=1}^k|Bb_r|^2,
- |Bf|^2le d|f|_∞^2|B|_{mathscr E}scrE^2,
+ |Bf|^2le d|f|_∞^2|B|_{E},
the second by testing B^*B against the positive matrix ff^*. Source
-conjugation mathscr U_sB=scrU_sB=widetilde U_sBU_s^* is isometric because
-sigma_s preserves tau_{mathcal N}calN, and base defects cancel since B
+conjugation U_sB= U_sBU_s^* is isometric because
+sigma_s preserves tau_{N}, 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;scrE; the
+|B_n|_{op}le|B|_{op} converge in E; the
bounded-column estimate gives strong convergence on bounded columns, and
B_nto B_∞ with B_∞ intertwining base and target and S-fixed.
-Then umapsto B_∞^*M_{u(l)}B_∞ is a nonzero positive full-flag
-map into mathcal A_j,calA_j, target-equivariant and S-invariant, contradicting
+Then u|-> B_∞^*M_{u(l)}B_∞ is a nonzero positive full-flag
+map into A_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,
@@ -2459,7 +2459,7 @@ unchanged image marginal. Set T^{(s)}=widetilde U_{b_s}^*TU_{b_s}. The
marginal lift fixes (x,l), so its fibre expression is
(U_{b_s}^X)^*D_x(l)^{1/2}U_{b_s}^X applied to Vf; inequality
(eq:links-positive-window) persists. With J_0f(x,l)=Vf(x),
-Bmapsto< Bf,J_0f> is continuous on mathscr EscrE by
+B|->< Bf,J_0f> is continuous on E by
(eq:links-intertwiner-frame). Weak decay and the positive inequality
give
0le√(c_C)< f,Psi_s(C)f>le< T^{(s)}f,J_0f>to0;
@@ -2517,7 +2517,7 @@ 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.bbRv.
+nonzero, since the common kernel of Pi∖{j} is Rv.
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
@@ -2540,7 +2540,7 @@ 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), with index product qp,
where A_0,B_0 are operator-bounded with Fourier expansions
@@ -2603,8 +2603,8 @@ ambient product and no discreteness of a proper projection.
We need Lipschitz stability of interpolation: from a fixed endpoint,
multiply the simple coordinates L_i of a dominant vector by t_iin[0,1].
In an archimedean polar chart, the angular differential in a positive-root
-direction is multiplied by sinh(alpha(tcdot L))/sinh(alpha(L))le1
-and the radial differential by the bounded linear map Lmapsto tcdot L;
+direction is multiplied by sinh(alpha(t· L))/sinh(alpha(L))le1
+and the radial differential by the bounded linear map L|-> t· 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
@@ -2646,12 +2646,12 @@ 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'|-> kp' and
+q^{-1}|-> k^{-1}q^{-1}. The reverse vertex of the first multiplier is
in V^+ (transverse to the entire buffered positive input patch); of the
second, in U^+ (transverse to the negative input patch). For a fixed
finite-dimensional representation, put S(g)=R(g)/|R(g)|. On either
-substitution rmapsto ur, transverse Cartan addition plus highest-weight
+substitution r|-> 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
@@ -2737,7 +2737,7 @@ in the three-slot system.
Proof.
Omit O first. Source covariance makes the uncut range join a projection
-over (mathcal A_1)^calA_1)^sigma; target covariance transports its range by
+over (A_1)^; 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
@@ -2839,7 +2839,7 @@ different candidate families at the fixed isolated type, there is no
common nonzero target-equivariant positive map dominated by the two
candidate maps (complete-positive domination). After induction, there is
no common nonzero linear or conjugate-linear functional on
-C^∞(Z)odotmathcal HcalH bounded by both angular form norms
+C^∞(Z)odotH bounded by both angular form norms
b⟼(int_Z< b(v),Phi_x(dv)b(v)>)^{1/2}.
@@ -2907,7 +2907,7 @@ 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 HcalH for disjoint angular patches
+sharp, the ranges P_x(U)P_y(V)H 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,
@@ -2976,7 +2976,7 @@ Let zsim F mean z is incident to F. Fix a positive stratum of a
mixed source face path or a pure vertex path F; denote its normalized
image-face marginal by Psi_F and the scalar mass of the unnormalized
stratum by c_F>0. For an open patch O of target type p(i), put
-mathcal I_i(O)=calI_i(O)={x:the target face x admits incident
+I_i(O)={x:the target face x admits incident
type-p(i) vertex in O}. For an archimedean highest-line module
V, let R_x be the orthogonal projector onto
span{n(v):vsim x, type(v)=p(i)}, and
@@ -2990,7 +2990,7 @@ define
The following inclusions hold:
[label=(roman*)]
- The support space of Psi_F(mathcal I_i(O))calI_i(O)) is contained in
+ The support space of Psi_F(I_i(O)) is contained in
overline{sum_{zsim F}ranPhi_z(O)}.
If the path is mixed and p(i) is archimedean, in every even highest
power the support of int R_x ⊗ Psi_F(dx) is contained in
@@ -3006,12 +3006,12 @@ vector in the stated closed end range.
Proof.
-We prove (i) by annihilating the common kernel mathcal C=∩_calC=∩_{zsim
-F}kerPhi_z(O). For mixed paths take any finmathcal CcalC and
+We prove (i) by annihilating the common kernel C=∩_calC=∩_{zsim
+F}kerPhi_z(O). For mixed paths take any finC and
operator-bounded f_lto f. For nonmixed paths supported in G_0 with
-N_0=mathcal N^calN^{G_0}: if i is in another component, ordinary fullness
-gives mathcal C=0calC=0 and the inclusion is immediate; otherwise
-Lemma~lem:links-relative-boundedness(ii) makes mathcal CcalC
+N_0=N^{G_0}: if i is in another component, ordinary fullness
+gives C=0 and the inclusion is immediate; otherwise
+Lemma~lem:links-relative-boundedness(ii) makes C
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.
@@ -3047,10 +3047,10 @@ transverse open cell; fullness and strong continuity make these end ranges
total, giving annihilation on the window for every end vector.
For the frame step: exhaust the compact windows to the event A of the
-chosen gap stratum and image condition mathcal I_i(O);calI_i(O); monotone
+chosen gap stratum and image condition I_i(O); monotone
continuity gives E(A)J(barbar b,f)=0.b,f)=0. Then A is invariant under ignored
left Fourier shifts, and the frame identity
-c_F< f,Psi_F(mathcal I_i(O))f>=calI_i(O))f>=frac1dsum_r
+c_F< f,Psi_F(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.
@@ -3176,9 +3176,9 @@ uniqueness, component by component.
For a general mixed face path of type I: the type-i source vertex z_i
incident to F is unique for iin I. Proposition~prop:transfer-incidence(i)
-puts Psi_F(mathcal I_i(O))calI_i(O)) inside P_{z_i}(O). A target face omitting
+puts Psi_F(I_i(O)) inside P_{z_i}(O). A target face omitting
p(i) admits two distinct incident vertices of that type, so disjoint
-O,O' make Psi_F(mathcal I_i(O)calI_i(O)capmathcal I_i(O'))calI_i(O')) supported in
+O,O' make Psi_F(I_i(O) I_i(O'))calI_i(O')) supported in
both P_{z_i}(O) and P_{z_i}(O'), hence zero. Countably many such
pairs exhaust the omitted-type event, so every p(i) for iin I
occurs. The upper bound of Proposition~prop:links-gap-count forces
@@ -3229,7 +3229,7 @@ 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 zmapsto sz under simultaneous
+common source parameter transports by z|-> 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
@@ -3376,11 +3376,11 @@ root-generated group is trivial.
There is an actual source Levi cover widetilde Sto S acting strongly
continuously on the endpoint spaces, commuting with the sharp
-endpoint-face observations, moving residue labels by zmapsto sz, and
+endpoint-face observations, moving residue labels by z|-> sz, and
its ineffective finite center acts trivially on the residue algebras. Our
goal is a fixed homogeneous analytic homeomorphism tau:B_Sto B_T and
measurable target projectivity observations a,b so that the endpoint
-families are evaluations of zmapsto atau(z) and zmapsto btau(z).
+families are evaluations of z|-> atau(z) and z|-> btau(z).
## The paired endpoint algebra
@@ -3404,7 +3404,7 @@ 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.bbC1.
+D_*^T=C1.
Proof.
@@ -3413,7 +3413,7 @@ 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
+overlinexi ⊗ eta|->|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,
@@ -3425,7 +3425,7 @@ Varying zeta, these intertwiners force the fibres d_t^+ to be graded
by the normal representation types: each Schur subspace remains
finite-dimensional per coset, giving a countable direct sum of finite
matrix algebras. Ergodicity of T acting transitively on Ubackslash
-T forces D_*^T=mathbb C1:bbC1: any T-fixed normal functional on the
+T forces D_*^T=C1:bbC1: any T-fixed normal functional on the
countable Cartesian product vanishes off the identity coset, and the
resulting character is scalar.
@@ -3466,7 +3466,7 @@ 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, D_0^U=mathbb Ce.bbCe.
+ D_0=eD_*e, D_0^U=Ce.
Choose decreasing compact open normal subgroups U_n△left U with
@@ -3556,7 +3556,7 @@ Hence:
[Dimensions and abelian corner]
-dim_{mathbb Q_p}bbQ_pmathfrak s=fraks=dim_{mathbb Q_p}bbQ_pmathfrak t.frakt. Every
+dim_{Q_p} s=fraks=dim_{Q_p} t.frakt. Every
Hausdorff limit G_∞ has both projections isomorphisms, and the
target-only subgroup G_∞cap(1times U) is {1}. Consequently
D_0 is abelian and equals L^∞(U) with Haar probability.
@@ -3595,14 +3595,14 @@ 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_*~= L^∞(T) with
left translation. In these coordinates, the commuting source action is
(beta_s f)(x)=f(xeta(s)), eta:Sto T,
where eta is a continuous homomorphism with isomorphic
-mathbb Q_p-differential.bbQ_p-differential.
+Q_p-differential.bbQ_p-differential.
Proof.
@@ -3620,24 +3620,24 @@ Countably many T-translates of e have join one, so D_* is type I.
Its center has a positive-measure U-invariant chart isomorphic to U,
and the T-action is ergodic (Equation~(eq:finite-fixed)).
Use Mackey's Borel point realization [cite]: let X realize
-the center, let varphi:Uto X be measurable onto the chart.
+the center, let phi:Uto X be measurable onto the chart.
Fubini in the two Haar variables gives x_0 with
-varphi(u)=ux_0 almost everywhere; for a generic u_0,
-varphi(vu_0)=vvarphi(u_0) for almost every v. Let
+phi(u)=ux_0 almost everywhere; for a generic u_0,
+phi(vu_0)=vphi(u_0) for almost every v. Let
L=Stab_T(x_0). If hin Lcap U, comparing u and
uh on a common conull set forces h=1, so Lcap U meets only the
identity in a small neighborhood, hence L is discrete and closed. The
orbit map T/Lto X is a Borel bijection with Borel inverse, countably
many translates of the chart cover modulo null sets, and
-Z(D_*)cong L^∞(T/L).
+Z(D_*)~= L^∞(T/L).
The measurable automorphisms of T/L commuting with left translation
are the right translations by N_T(L)/L (explicitly: for such F,
-gmapsto g^{-1}F(gL) is left-invariant, hence constant =nL, giving
+g|-> g^{-1}F(gL) is left-invariant, hence constant =nL, giving
n^{-1}Lnsubset L, equality by applying to F^{-1}). The injection
N_T(L)/LtoAut(T/L) has a Borel inverse. The source
action gives a measurable (hence continuous, analytic over
-mathbb Q_p)bbQ_p) homomorphism eta:Sto N_T(L)/L.
+Q_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)
@@ -3650,15 +3650,15 @@ open subgroup; the open-centralizer assertion forces l=1. Thus
L={1} and Z(D_*)=L^∞(T) with left translation.
Remove matrix fibres: the type I decomposition and ergodicity of the
-center give D_*cong L^∞(T;B(mathcal V))calV)) with constant separable
+center give D_*~= L^∞(T;B(V)) with constant separable
dimension. In a measurable trivialization the action is
(alpha_ad)(x)=c(a,x)d(a^{-1}x) with
-c(ab,x)=c(a,x)c(b,a^{-1}x) (an Aut(B(mathcal V))-calV))-
+c(ab,x)=c(a,x)c(b,a^{-1}x) (an Aut(B(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);calV); Equation~(eq:finite-fixed) forces dimmathcal
+B(V); Equation~(eq:finite-fixed) forces dimmathcal
V=1.calV=1.
@@ -3702,11 +3702,11 @@ and Fubini give F(x,z)=xF_0(z), and the source action gives
in the ordinary measure class on B_S.
Regard each effective adjoint group over its coefficient field as a
-restriction of scalars to mathbb Q_p;bbQ_p; in characteristic zero, all
+restriction of scalars to Q_p; in characteristic zero, all
derivations of a semisimple Lie algebra are inner, and the identity
component of its algebraic automorphism group is the adjoint algebraic
group. Conjugation by A=deta is an isomorphism of these adjoint
-algebraic groups over mathbb Q_p,bbQ_p, and
+algebraic groups over Q_p, and
AAd(s)=Ad(eta(s))A, sin S.
@@ -3718,7 +3718,7 @@ the descent condition identifies with a single parabolic over the
original field); thus tau:B_Sto B_T exists, intertwining S with
eta(S).
-Write B_S=S/P; the measurable lift smapstoeta(s)^{-1}F_0(sP) is
+Write B_S=S/P; the measurable lift s|->eta(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
@@ -3749,7 +3749,7 @@ injective, with Borel inverse.
## Uniform finite-place upper speed
For each finite-place effective panel group R, choose a norm on its
-prime-field adjoint Lie algebra, with |p|=p^{-1} on mathbb Q_p,bbQ_p, and
+prime-field adjoint Lie algebra, with |p|=p^{-1} on Q_p, and
let ell_R be its rank-one Cartan gap. Constants d_R>0 depend only
on the panel type:
@@ -3758,9 +3758,9 @@ on the panel type:
with uniform error. On a dominant split translation the largest adjoint
eigenvalue has logarithm a fixed positive multiple of the simple-root
-coordinate; converting from |cdot|_k to |cdot|_p^{[k:mathbb Q_p]}bbQ_p]
+coordinate; converting from |·|_k to |·|_p^{[k: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
+d_R=r_{max}/[k: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.
@@ -3938,11 +3938,11 @@ 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 VcalV
-induced by the surjective bracket ∧^2mathcal VcalVto
-T∂ X/mathcal V.calV. It is compact-invariant with mass one. Naturality
+horizontal volume density with the density on T∂ X/V
+induced by the surjective bracket ∧^2V
+T∂ X/V. It is compact-invariant with mass one. Naturality
of the bracket makes it the compact angular probability, and its Jacobian
-under a horizontal dilation dcdotIsom is exactly d^Q.
+under a horizontal dilation d·Isom is exactly d^Q.
[Triangle defect]
There is C such that if x,yin X and
@@ -3990,7 +3990,7 @@ by the factorization.
[Radial cutoffs]
Let fin C^∞([0,∞)) be constant near zero and near infinity.
-There are R_f,C_f>0 such that (x,x')mapsto f(d(x,x')/R) has Schur
+There are R_f,C_f>0 such that (x,x')|-> 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
@@ -4043,17 +4043,17 @@ unchanged, giving complete boundedness.
## One-sided localization
Let D=Bbarbar ⊗⊗ L_eta(Gamma) with product trace tau and
-Fourier coefficients Y_hin L^2(B); D_varphi denotes coefficient
+Fourier coefficients Y_hin L^2(B); D_phi denotes coefficient
multiplication on L^2(D). For finite Fourier sums,
(YY^*)_g=sum_heta(gh,h^{-1})overline{eta(h,h^{-1})}Y_{gh}Y_h^*,
and phases have absolute value one.
[Unary Plancherel estimate]
-Let varphi_{i,nu}:Gammatomathbb CbbC satisfy
-sup_hsum_i|varphi_{i,nu}(h)|^2le b for every nu. With T_g an
+Let phi_{i,nu}:GammatoC satisfy
+sup_hsum_i|phi_{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
+z_{g,t}=sum_nu c_{g,t,nu}[Y(D_{phi_{i(g,t),nu}}Y)^*]_g where
|c_{g,t,nu}|le Aa_nu, sum_nu a_nu<∞, we have
(sum_gsum_{tin T_g}|z_{g,t}|_{2,B}^2)^{1/2}
@@ -4065,7 +4065,7 @@ The same holds with the first factor localized instead.
Proof.
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
+sum_{g,t}|[Y(D_{phi_{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.
@@ -4141,10 +4141,10 @@ finishes.
## The ray lift and its densities
-Let mathcal K_X=L^2(X,dp)calK_X=L^2(X,dp) and let T_g be translation on
-mathcal K_X.calK_X. Define
+Let K_X=L^2(X,dp)calK_X=L^2(X,dp) and let T_g be translation on
+K_X. Define
[
- Delta:D→ B(mathcal K_X)calK_X)barbar ⊗⊗ D,
+ Delta:D→ B(K_X) ⊗⊗ D,
Delta(bu_h)=T_h ⊗ bu_h,
]
a normal unital homomorphism (conjugate by the unitary sending
@@ -4162,9 +4162,9 @@ ideal endpoint v, use a chart sending v to infinity and put
This is equivariant in x,p,v and has unit L^2 norm, supported in a
bounded neighborhood of the ray from x to v at depths
[d_1-O(1),2d_1+O(1)]. The coefficient isometry is
-mathbb V_bbV_{d_1}Y=sum_htheta_{o,d_1}(cdot,h^+) ⊗ Y_hu_h.
+V_{d_1}Y=sum_htheta_{o,d_1}(·,h^+) ⊗ Y_hu_h.
-For a column xiinmathcal K_XcalK_X ⊗ L^2(D), the left and right
+For a column xiinK_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),
@@ -4176,21 +4176,21 @@ countable homomorphisms to Isom(X), with
d_1to∞, l/d_1to∞, and Yin D Fourier-supported in
{rge l}.
[label=(roman*)]
- For large indices, mathbb V_bbV_{d_1}Y has left density for
+ For large indices, 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_bbV_{d_1}Y(mathbb V_bbV_{d_1}Y)^*|_{2,Tr ⊗ tau} has this
+|V_{d_1}Y(V_{d_1}Y)^*|_{2,Tr ⊗ tau} has this
bound.
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.
+|V_{d_1}(xY)-Delta(x)V_{d_1}Y|_2to0 and
+|V_{d_1}(Yx)-(V_{d_1}Y)x|_2to0.
For another long column Z and bounded x, if for every
contributing triple hinsuppwidehatwidehat Y,Y, hkinsuppwidehatwidehat Z,Z,
kinsuppwidehatwidehat xx one has
d_{∂ X}(h^+,(hk)^+)le e^{-cl'} with c>0,
l'/d_1to∞, then
- <mathbb V_bbV_{d_1}Z,(mathbb V_bbV_{d_1}Y)x>
+ <V_{d_1}Z,(V_{d_1}Y)x>
-< Z,Yx>→0,
uniformly with |x|_∞,|Y|_2,|Z|_2 bounded.
@@ -4216,8 +4216,8 @@ 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_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
+|V_{d_1}Y(V_{d_1}Y)^*|_2^2
+=|(V_{d_1}Y)^*V_{d_1}Y|_2^2 and monotone
approximation.
@@ -4228,7 +4228,7 @@ There are smooth annular partitions about g^+ at integer depths
0le jle L (with exterior piece and innermost ball) on which
- |w_{d_1}(g,cdot)-a_{d_1}(j,L)|_{C^m_{rm normalized}}
+ |w_{d_1}(g,·)-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,
@@ -4299,7 +4299,7 @@ a=o(d_1) gives difference bounded in L^2 by
Polarization extends to
- <mathbb V_bbV_{d_1}Z,Delta(x)mathbb V_bbV_{d_1}Y>
+ <V_{d_1}Z,Delta(x)V_{d_1}Y>
-< Z,xY>→0
uniformly when norms of Z,Y,x are bounded with
@@ -4325,7 +4325,7 @@ matrix norm times the frequency sum vanishes. This proves
r(h),r(hk)ge l-a and r(k)le a give
d_{∂ X}(h^+,(hk)^+)le Ce^{-c(l-2a)} with fixed c>0, so
(iii) applies. Expanding
-|mathbb V_bbV_{d_1}(Yx)-(mathbb V_bbV_{d_1}Y)x|_2^2 finishes (ii) for
+|V_{d_1}(Yx)-(V_{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.
@@ -4334,8 +4334,8 @@ and Z=xY.
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
+|Delta(x-x')V_{d_1}Y|_2to0 and
+|(V_{d_1}Y)(x-x')|_2to0; the analogous amplified
statement holds in the semifinite L^2 norm.
@@ -4343,11 +4343,11 @@ 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_bbV_{d_1}Y|_2^2,
- |(mathbb V_bbV_{d_1}Y)z|_2^2}
+ max{|Delta(z)V_{d_1}Y|_2^2,
+ |(V_{d_1}Y)z|_2^2}
le C|Y|_∞|Y|_2|z|_∞|z|_2.
-The amplified version uses mathbb V_bbV_{d_1}Y(mathbb V_bbV_{d_1}Y)^*
+The amplified version uses V_{d_1}Y(V_{d_1}Y)^*
and the semifinite norm. For fields, integrate the pointwise bound
and apply Cauchy--Schwarz on the base.
@@ -4362,8 +4362,8 @@ with symbol f. Finite factor traces are normalized.
An admissible representation for T is a finite factor
D=Bbarbar ⊗⊗ L_eta(Gamma) together with a normalized
-projective representation Θ:Gammatomathcal U(D)calU(D) such that
-DcapΘ(Gamma)'=mathbb C1bbC1 and the representation
+projective representation Θ:GammatoU(D) such that
+DcapΘ(Gamma)'=C1 and the representation
gamma⟼AdDelta_T(Θ(gamma))
quadhbox{on}
@@ -4460,14 +4460,14 @@ 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}·),
the mixed Gram forms are pairings between A_{n,f'}^*A_{n,f} and
the translate of B_{n,g}B_{n,g'}^* under the induced trace action.
Its invariant space consists of scalars; averaging with m_0^{*k}
gives a uniform O(c_0^k) gap. Two successive Hilbert limits give an
isometry
- J:L^2(P_t) ⊗ L^2(P_u)→mathcal K,calK,
+ J:L^2(P_t) ⊗ L^2(P_u)→K,
f ⊗ g⟼(A_{n,f}(sy)X^{Θ_n}_s(y)B_{n,g}(y))^{lim}.
@@ -4478,7 +4478,7 @@ A(sy)^* pairs against X^{Θ_n}_s(y)B(y)C(y,s); convert to
tau(zeta_n(sy)^*Delta_T(X^{Θ_n}_s(y))xi_n(y)C(y,s))
-with xi_n=mathbb V_bbV_{d_n}B, zeta_n=mathbb V_bbV_{d_n}(A^*). The
+with xi_n=V_{d_n}B, zeta_n=V_{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
@@ -4506,7 +4506,7 @@ K_j↑ F^c gives E(K_j)JP_{rm in}(a^{-1}(F))=0, so
E(F^c)JP_{rm in}(a^{-1}(F))=0; together,
E(F)J=J P_{rm in}(a^{-1}(F)). Monotone classes and spectral
integration give
-E(varphi)J=JM_{varphi(s+t)} for varphiin C([0,∞]).
+E(phi)J=JM_{phi(s+t)} for phiin C([0,∞]).
Convergence on the whole sequence follows by compactness of
probability measures on [0,∞].
@@ -4515,11 +4515,11 @@ probability measures on [0,∞].
At an isolated type, Section~sec:transfer has already supplied
-sharp boundary measurements xmapsto P_x. We now construct a height
+sharp boundary measurements x|-> 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 xmapsto P_x; the main analytic requirement
+horizontal derivative of x|-> 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,
@@ -4566,7 +4566,7 @@ not new assumptions.
For the two-slot model of Section~sec:setup there is a finite
tensor D=Bbarbar ⊗⊗ L_eta(Gamma), with B diffuse, and an
-admissible representation delta:Gammatomathcal U(D)calU(D) for each
+admissible representation delta:GammatoU(D) for each
isolated measured factor, obtained by a fixed finite matrix corner
equivalence. It has a sequence gamma_n whose Fourier laws escape
in every measured simple factor. Rescaled continuous radial laws,
@@ -4668,17 +4668,17 @@ argument finishes.
For cutoffs L_nge1, a bounded sequence x_nin D_n is
tight at scale L_n if
- lim_{Cto∞}lim_{ntomathcal U}calU
+ lim_{Cto∞}lim_{ntoU}
|D_{1_{{r_T>CL_n}}}x_n|_2=0
-for a free ultrafilter mathcal U.calU. Ultraproducts are tracial;
+for a free ultrafilter U. Ultraproducts are tracial;
ordinary subsequences are extracted once all countably many
requirements are fixed.
The tight elements form a von Neumann subalgebra Q_0 of
-prod_{mathcal U}calUD_n, and Q_0 is a factor. If each Θ_n
+prod_{U}, and Q_0 is a factor. If each Θ_n
is relatively irreducible, the ultraproduct representation Θ
also has scalar relative commutant.
@@ -4707,12 +4707,12 @@ Approximate commutation forces z scalar.
## Near-identity Kazhdan tightening
[Kazhdan tightening]
-Fix a finite Kazhdan set mathcal FcalFsubsetGamma and
+Fix a finite Kazhdan set F and
ε>0. There is ε_K>0 such that, if
Θ_n are admissible and
|D_{1_{{r_T>L_n}}}Θ_n(gamma)|_2leε_K
- (gammainmathcal F),calF),
+ (gammainF),
then, after passing to a subsequence, there exist unitaries
@@ -4722,9 +4722,9 @@ Admissibility is preserved.
Proof. [Sketch]
-Let D_mathcal U=calU=prod_{mathcal U}calUD_n and Q_0 the tight
+Let D_U={U} and Q_0 the tight
factor of Lemma~lem:scale-tight-factor. The Jones basic
-construction < D_mathcal U,e_calU,e_{Q_0}> gives
+construction < D_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,
@@ -4740,7 +4740,7 @@ e_{Q_0}; spectral rounding p_0=1_{[1/2,1]}(z) gives
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}
+take a right Q_0-module isometry with row Vin L^2(D_U)^{ ⊕ 2}
satisfying Vq=V, E_{M_2(Q_0)}(V^*V)=q, Θ(gamma)V=Vb_gamma;
VV^*=d'1.
@@ -4758,7 +4758,7 @@ q,b_gamma with support r_Tle a_n:
|G_n-H_{1,n}|_2to0.
-The ray lift sends G_n to xi_n=mathbb V_bbV_{d_n}G_n with
+The ray lift sends G_n to xi_n=V_{d_n}G_n with
Delta_T(b_{gamma,n})xi_n-xi_nb_{gamma,n}to0,
Delta_T(q_n)xi_n-xi_nto0,quadxi_nq_n-xi_nto0
@@ -4826,7 +4826,7 @@ formula
composes the kernels. Factoriality and admissibility come from
the induced coefficient representation
- (mathcal U_gk)(y)=calU_gk)(y)=Ad T_g(y),k(g^{-1}y),
+ (U_gk)(y)= T_g(y),k(g^{-1}y),
with scalar invariant space; for bounded finite tests and
@@ -4834,12 +4834,12 @@ 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_icalU_i k>|,
+ |< k',U_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 EcalE by
+The amplified assertion replaces E 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.
@@ -4865,7 +4865,7 @@ limit P^{*m}*Q with P([a,∞])gealpha and
Q supported in [0,∞]. For finite M,
(P^{*m}*Q)([0,M])le
- mathbb PbbP{Bin(m,alpha)le M/a}to0
+ P{Bin(m,alpha)le M/a}to0
as mto∞; diagonal extraction over countably many m
@@ -4876,7 +4876,7 @@ Lemma~lem:ray-compactshort.
[Positive mass at linear speed]
For each isolated source type i, matched to p(i), there
exist C<∞, alpha>0, T_nto∞ such that for
-every fixed xiinmathcal H,calH,
+every fixed xiinH,
liminf_{n}inf_{T_n/2le tle T_n}
bigl|D_{1_{{r_{p(i)}le CT_n}}}U_{a_t}xibigr|^2
@@ -4892,11 +4892,11 @@ Proof. [Sketch]
Use delta from Lemma~lem:scale-original-model.
Suppose the matched length is totally superlinear:
- P_{delta,t}(t)([0,C])to0qquad(C<∞).
+ P_{delta,t}(t)([0,C])to0 (C<∞).
Compose the original representation around the phase cycle
-imapsto p(i)mapsto...mapsto i via
+i|-> p(i)|->...|-> 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
@@ -4908,7 +4908,7 @@ D'_nto∞: feasibility is nonempty (identity embedding
into L_eta(Gamma) suffices), so the infimum
inf{tge1:|D_{1_{{r_i>L}}}Θ(gamma)|_2leε_K
- (gammainmathcal F),calF), P_{Θ,D'_nL}(t)([1,∞])getfrac12}
+ (gammainF), P_{Θ,D'_nL}(t)([1,∞])getfrac12}
is attained up to factor two at some witness. Lemma~lem:scale-tightening
@@ -4951,13 +4951,13 @@ Hilbert density extend the bound to every fixed xi.
Fix matched isolated factors S,T with boundaries B_S,B_T
homogeneous dimensions Q_S,Q_T, Carnot distances d_S,d_T, compact
-angular probability dx, and horizontal bundles mathcal V_R.calV_R. The
+angular probability dx, and horizontal bundles V_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 EscrEsubsetmathcal HcalH
+All isolated factors here have Q_T>2. Let E HcalH
be the maximal invariant linear domain of
Proposition~prop:scale-pair-estimate, on which
(eq:input-10) holds uniformly over each fixed collection of
@@ -4965,10 +4965,10 @@ normalized mask controls.
For Fin C^∞(B_Ttimes B_T), its ordered evaluation is
-mathcal K_calK_{x,y}(F)=iint F(w,v),P_y(dw)P_x(dv) (finite separated
+K_{x,y}(F)=iint F(w,v),P_y(dw)P_x(dv) (finite separated
decomposition on rectangles). A smooth kernel has horizontal order at
least one if F|_{w=v}=0; at least two if in addition
-d_zF|_{mathcal V_T}calV_T vanishes on the diagonal.
+d_zF|_{V_T} vanishes on the diagonal.
@@ -4977,8 +4977,8 @@ 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*)]
- Order at least one gives, for each xiinmathscr E,scrE,
-|mathcal K_calK_{x,y}(F)xi|_{L^2(dnu_ε;mathcal H)}calH)
+ Order at least one gives, for each xiinE,
+|K_{x,y}(F)xi|_{L^2(dnu_ε;H)}
le C_{F,xi}ε^{Q_S/Q_T}.
If Q_S=Q_T=Q>2 and order is at least two, the same norm is
o(ε), explicitly bounded by
@@ -4988,7 +4988,7 @@ o(ε), explicitly bounded by
Proof.
-The measure pullback under wmapsto y has density at most
+The measure pullback under w|-> 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
@@ -5014,9 +5014,9 @@ listed.
Every matched isolated pair has Q_S=Q_T=Q. For every smooth
horizontal source field X, every fin C^∞(B_T), and every
-xiinmathscr E,scrE, the distributional derivative
-D_X(f)xi=X(xmapsto P_x(f)xi) lies in
-L^2(B_S,dx;mathcal H),calH), linearly in f,xi and in smooth source
+xiinE, the distributional derivative
+D_X(f)xi=X(x|-> P_x(f)xi) lies in
+L^2(B_S,dx;H), linearly in f,xi and in smooth source
function multiples of X.
@@ -5026,7 +5026,7 @@ horizontal weight psi_i with first moments the coordinate vector
gives difference quotients
ε^{-1}intpsi_i(w)(P_{xdelta_ε w}(f)-P_x(f))xi,dw
bounded in L^2 by Cε^{Q_S/Q_T-1}; adjoint pairing recovers
-the distributional derivative -Xvarphi-(divX)varphi.
+the distributional derivative -Xphi-(divX)phi.
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
@@ -5036,18 +5036,18 @@ give D_X(f)xiin L^2 on charts; a partition of unity globalizes.
-Set mathcal K=L^2(B_S,dx;calK=L^2(B_S,dx;mathcal H)calH) and
-mathcal C=spancalC=span{xmapsto a(x)P_x(g)xi:
-ain L^∞(B_S), gin C^∞(B_T), xiinmathscr EscrE}. For real
+Set K=L^2(B_S,dx;calK=L^2(B_S,dx;H) and
+C=span{x|-> a(x)P_x(g)xi:
+ain L^∞(B_S), gin C^∞(B_T), xiinE}. For real
f and real horizontal X, the rule
T_X(f)(sum_ja_jP_bullet(g_j)xi_j)
=sum_ja_jP_bullet(g_j)D_X(f)xi_j
-defines a densely defined symmetric operator on mathcal C.calC. If real
+defines a densely defined symmetric operator on C. If real
smooth g_j,f_j satisfy sum_jg_j,d_{rm hor}f_j=0, then
- sum_jP_bullet(g_j)T_X(f_j)=0quadhbox{on }mathcal C.calC.
+ sum_jP_bullet(g_j)T_X(f_j)=0quadhbox{on }C.
@@ -5055,7 +5055,7 @@ Proof.
Write Delta_{x,y}f=P_y(f)-P_x(f). Sharpness gives the exact
identity
(Delta_{x,y}f)P_x(g)-P_y(g)Delta_{x,y}f
-=mathcal K_calK_{x,y}((f(w)-f(v))(g(v)-g(w))); the right side has
+=K_{x,y}((f(w)-f(v))(g(v)-g(w))); the right side has
horizontal order two and vanishes in the recovered L^2 average
after dividing by ε. Move Delta_{x,y}f by adjunction and
replace P_y(g) by P_x(g) (dominated convergence); the resulting
@@ -5066,7 +5066,7 @@ derivative of the difference quotients is
for almost every x. Inserting scalar parameter tests gives the
pointwise form; complex weights follow by linearity. Substituting
-finitely many angular tests into a formal pairing of u,vinmathcal CcalC
+finitely many angular tests into a formal pairing of u,vinC
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
@@ -5081,11 +5081,11 @@ theta=sum_jg_jd_{rm hor}f_j, set T_X(theta)=sum_j
P_bullet(g_j)T_X(f_j); equation (eq:scale-horizontal-cancellation)
makes this independent of expression. (Global finitely many target
coordinate differentials span, and their Gram operator is smoothly
-invertible on mathcal V_T.)calV_T.)
+invertible on V_T.)
There is a positive self-adjoint measurable field l_x on
-mathcal H,calH, affiliated to mathcal A_1,calA_1, strongly commuting with
+H, affiliated to A_1, strongly commuting with
P_x and invariant under complementary source factors, such that
sigma_a(l_x)=dil(a,x),l_{ax},
@@ -5097,24 +5097,24 @@ and l is not identically zero.
Proof.
Choose finite smooth Parseval horizontal source fields (X_alpha)
and target covectors (theta_r) (invert positive Gram operators on
-mathcal V_R).calV_R). Define the finite column
-D:mathcal CcalCtomathcal K^calK^{n_Sn_T},
+V_R). Define the finite column
+D:C K^calK^{n_Sn_T},
Du=(T_{X_alpha}(theta_r)u)_{alpha,r}, closable by
-mathcal CcalC density; set l=|overlineoverline D|.D|. Parameter and angular
-unitary multipliers commute with D on mathcal C,calC, extending to
+C density; set l=|overlineoverline D|.D|. Parameter and angular
+unitary multipliers commute with D on C, extending to
the closed graph and form, so l=int^{ ⊕ }l_x,dx with l_x
strongly commuting with P_x. Unitary invariance transfers
distributionally to D_X(f) and then closes, giving affiliation and
complementary invariance.
-For covariance, note that Parseval systems embed mathcal V_S,calV_S,mathcal V_TcalV_T
+For covariance, note that Parseval systems embed V_S,V_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;calC;
+(eq:scale-derivative-modulus-covariance) holds on C;
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
@@ -5125,7 +5125,7 @@ would make P_x constant, contradiction.
[Height at an isolated type]
For matched isolated pairs Q_S=Q_T=Q. There are positive injective
-self-adjoint b_x, affiliated to mathcal A_1,calA_1, strongly commuting
+self-adjoint b_x, affiliated to A_1, strongly commuting
with P_x and invariant under complementary source factors, with
sigma_a(b_x)=dil(a,x)^{-Q}b_{ax},
@@ -5136,12 +5136,12 @@ The construction passes to normal copies used in amplified models.
Proof.
Make l_x everywhere covariant: at x_0in B_S with parabolic
-stabilizer J, pull back along gmapsto gx_0 and set
+stabilizer J, pull back along g|-> gx_0 and set
F(g)=dil(g,x_0)sigma_g^{-1}(l_{gx_0}). Haar
disintegration over S/J and
(eq:scale-derivative-modulus-covariance) give F(ag)=F(g)
almost everywhere for each fixed ain S; Fubini over
-(a,g)mapsto(ag,g) reduces F to an essential value L_0 (test
+(a,g)|->(ag,g) reduces F to an essential value L_0 (test
via bounded resolvents and countably many matrix coefficients). For
kin J, F(gk)=dil(k,x_0)sigma_k^{-1}F(g), so
sigma_kL_0=dil(k,x_0)L_0, defining
@@ -5152,7 +5152,7 @@ and transports.
Let e_0=supp L_0. Target dilation is bounded, positive, invertible,
commuting with L_0; taking supports gives beta_h(e_0)=e_0, so
-e_0inmathcal N.calN. Source stabilizer covariance gives
+e_0inN. 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
@@ -5161,18 +5161,18 @@ 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;calA_1; taking inverse powers in the covariance laws and
+to A_1; taking inverse powers in the covariance laws and
using strong commutation with the dilation gives
(eq:scale-isolated-height-covariance).
-For amplification: on a normal copy mathcal HcalH ⊗ mathcal LcalL with
+For amplification: on a normal copy H ⊗ L with
orthonormal basis (e_n), an element Xi=sum_nxi_n ⊗ e_n of
the maximal amplified estimate domain has every coordinate
-xi_ninmathscr EscrE (each coordinate energy is bounded by the
-total); conversely finite sums of xi_ninmathscr EscrE lie in the
+xi_ninE (each coordinate energy is bounded by the
+total); conversely finite sums of xi_ninE lie in the
amplified domain. Finite-coordinate projections commute with the PVMs,
preserve the domain, and commute with weak derivatives; on the
-algebraic tensor core mathcal CcalCodotspan{e_n} the
+algebraic tensor core C{e_n} the
column is exactly D ⊗ 1, graph-densely. So
l^{rm amp}=l ⊗ 1 and b^{rm amp}=b ⊗ 1; unitary
identification with a normal copy preserves the identities, without