diff 56bc728..f6bf760 equivariant_jiang_su_stability_v2.txt
diff --git a/equivariant_jiang_su_stability_v2.txt b/equivariant_jiang_su_stability_v2.txt
index ef854c0..29a94b8 100644
--- a/equivariant_jiang_su_stability_v2.txt
+++ b/equivariant_jiang_su_stability_v2.txt
@@ -3,19 +3,19 @@ tensorially. The dynamic counterpart asks an action to absorb the trivial
action on Z up to cocycle conjugacy. The main theorem establishes this
for every amenable action in the simple, nuclear, stably finite, unital
setting. Throughout, C*-algebras are complex, tensor products are
-minimal, mathcal T(B)calT(B) is the tracial-state space of a unital C*-algebra
+minimal, T(B) is the tracial-state space of a unital C*-algebra
B, and Z is the Jiang-Su algebra.
The Jiang-Su algebra Z is the unique simple unital inductive limit of
prime dimension-drop intervals with the K-theory of the complex numbers and
a unique tracial state; it is strongly self-absorbing, so
-Z cong Z ⊗ Z. The dimension-drop construction and these properties
+Z ~= Z ⊗ Z. The dimension-drop construction and these properties
appear in the study of strongly self-absorbing algebras.
./pdf/toms-winter-math0502211-strongly-self-absorbing.pdf
Theorem. Let A be a simple, separable, unital, infinite-dimensional,
-nuclear, stably finite C*-algebra with A cong A ⊗ Z. Let G be a
+nuclear, stably finite C*-algebra with A ~= A ⊗ Z. Let G be a
countable discrete amenable group and alpha : G to Aut(A) an
action. Put B = A ⊗ Z and beta_g = alpha_g ⊗ id_Z.
Then there are a unital *-isomorphism Phi : A to B and unitaries
@@ -32,7 +32,7 @@ finite F subset G with |hF ,△, F| < delta |F| for every
h in H. The displayed relations say exactly that alpha and
beta = alpha ⊗ id_Z are cocycle conjugate, with
Phi intertwining the two actions through the beta-cocycle
-g mapsto u_g.
+g |-> u_g.
No outerness, Rokhlin, or Universal Coefficient Theorem assumption is made,
and no condition is placed on the action of G on the tracial boundary or
@@ -55,7 +55,7 @@ comparison of positive elements.
Theorem 1.1 (Matui and Sato). Let A be a unital separable simple
infinite-dimensional nuclear C*-algebra with finitely many extremal
-traces. Then the following are equivalent: A ⊗ Z cong A; A
+traces. Then the following are equivalent: A ⊗ Z ~= A; A
has strict comparison; every completely positive map from A to A
can be excised in small central sequences; has property (SI).
@@ -72,12 +72,12 @@ The forward implication is the absorption step; it remains to construct
the required invariant central trace splittings uniformly over the whole
trace space, even when the action moves traces.
-For a unital C*-algebra B and tau in mathcal T(B)calT(B) write
+For a unital C*-algebra B and tau in T(B) write
|x|_{2,tau} = tau(x^*x)^{1/2} and
-|x|_{2,u} = sup_{tau in mathcal T(B)}calT(B) |x|_{2,tau}. Under the
+|x|_{2,u} = sup_{tau in T(B)} |x|_{2,tau}. Under the
hypotheses on A, nuclearity implies exactness, and a unital stably
finite exact algebra carries a tracial state; every 2-quasitrace on a
-unital exact algebra is a trace. Hence mathcal T(A)calT(A) is a nonempty
+unital exact algebra is a trace. Hence T(A) is a nonempty
compact convex set, and each trace on the simple algebra A is faithful,
because its null space is a closed two-sided ideal not containing the
unit.
@@ -90,16 +90,16 @@ Corollary 5.12 (Haagerup). Every stably finite unital exact C*-algebra has a tra
Fix a free ultrafilter omega on the positive integers and form the
uniform tracial ultrapower
- A^omega = ell^∞(mathbb N,bbN, A) / I_omega, I_omega = {(a_j) : lim_{j to omega} |a_j|_{2,u} = 0}.
+ A^omega = ell^∞(N, A) / I_omega, I_omega = {(a_j) : lim_{j to omega} |a_j|_{2,u} = 0}.
The inequalities |ab|_{2,u} le |a| |b|_{2,u} and
|ba|_{2,u} le |a| |b|_{2,u} show that I_omega is a closed
two-sided ideal, and faithfulness of traces embeds A as the constant
-sequences. For any sequence (tau_j) in mathcal T(A)calT(A) the formula
+sequences. For any sequence (tau_j) in T(A) the formula
tau([(a_j)]) = lim_{j to omega} tau_j(a_j) defines a tracial state
on A^omega, because |tau_j(a_j)| le |a_j|_{2,tau_j}. These are
-the limit traces, written mathcal T_calT_omega(A); these need not exhaust
-the traces of A^omega. Automorphisms preserve |cdot|_{2,u}, and
+the limit traces, written T_; these need not exhaust
+the traces of A^omega. Automorphisms preserve |·|_{2,u}, and
applying alpha_g coordinatewise gives an action
baralpha : G curvearrowright A^omega preserving A^omega cap A'.
@@ -107,7 +107,7 @@ An action alpha : G curvearrowright A has equivariant uniform property
Gamma if for every integer k ge 2 there are pairwise orthogonal
projections p_1, ..., p_k in (A^omega cap A')^{baralpha} with
tau(a p_i) = tau(a)/k for every a in A, every
-tau in mathcal T_calT_omega(A), and every i. The construction also
+tau in T_, and every i. The construction also
forces sum_i p_i = 1. The absorption criterion to be used states that a
countable discrete amenable group acting on a separable unital simple
nuclear Jiang-Su stable C*-algebra with nonempty trace space, and having
@@ -131,15 +131,15 @@ commute with A and commute pairwise. Consequently
C = C^*(∪_{i ge 1} C_i) ⊆ A^omega cap A'
is a unital separable baralpha-invariant subalgebra with
-mathcal T(C)calT(C) neq ∅. There are moreover self-adjoint elements
+T(C) != ∅. There are moreover self-adjoint elements
x_{i,g} in C_i and a constant M < ∞ with
- |x_{i,g}| le M, varphi(x_{i,g}) = 0, varphi(x_{i,g}^2) = 1 (varphi in mathcal T(C)),calT(C)), baralpha_h(x_{i,g}) = x_{i,hg}
+ |x_{i,g}| le M, phi(x_{i,g}) = 0, phi(x_{i,g}^2) = 1 (phi in T(C)), baralpha_h(x_{i,g}) = x_{i,hg}
for all i ge 1 and g, h in G.
Proof. Since A is Z-stable, fix an identification
-A cong A ⊗ ⊗_{l ge 1} Z and use its increasing finite
+A ~= A ⊗ ⊗_{l ge 1} Z and use its increasing finite
tensor stages. First, every norm-separable C*-subalgebra
D ⊆ A^omega admits a unital embedding Z to A^omega cap D'.
Choose a norm-dense sequence (d_s)_{s ge 1} in D with bounded
@@ -151,7 +151,7 @@ factor strictly after that stage. For j ge s and z in Z,
|[theta_j(z), d_{s,j}]| le (2|z|)/(j).
The coordinatewise homomorphisms therefore induce a unital *-homomorphism
-z mapsto [(theta_j(z))_j] into A^omega cap D', injective because Z
+z |-> [(theta_j(z))_j] into A^omega cap D', injective because Z
is simple and the unit of A^omega is nonzero.
Set D_0 = A. Suppose D_{i-1} is a separable unital
@@ -166,13 +166,13 @@ g in G, invariance of D_{i-1} gives
so C_i commutes with D_{i-1}. Countability of G keeps each D_i
separable, yielding the required algebras and commutation. A limit trace on
A^omega restricts to a tracial state on C, so
-mathcal T(C)calT(C) neq ∅.
+T(C) != ∅.
The trace rho of Z is faithful by simplicity. Pick a non-scalar
self-adjoint c in Z; then rho((c - rho(c)1)^2) > 0, and rescaling
c - rho(c)1 gives a self-adjoint b in Z with rho(b) = 0 and
rho(b^2) = 1. Set x_{i,g} = baralpha_g(iota_i(b)) and
-M = |b|. Every varphi in mathcal T(C)calT(C) restricts along each
+M = |b|. Every phi in T(C) restricts along each
translated unital embedding of Z to rho, giving the normalization, and
the action law gives the equivariance. This proves the proposition.
@@ -180,51 +180,51 @@ The next fact is intrinsic to the algebra generated by the rows, so it
applies to traces of C that do not extend to the ambient ultrapower.
Lemma. Let B be a unital C*-algebra generated by pairwise commuting
-unital C*-subalgebras B_i, i ge 1. If varphi is an extreme point of
-mathcal T(B),calT(B), then
+unital C*-subalgebras B_i, i ge 1. If phi is an extreme point of
+T(B), then
- varphi(b_1 ... b_s) = prod_{j=1}^s varphi(b_j)
+ phi(b_1 ... b_s) = prod_{j=1}^s phi(b_j)
whenever b_j in B_{i_j} and the indices i_1, ..., i_s are distinct.
-Proof. Let (H_varphi, pi, Ω) be the tracial GNS representation and
-put mathcal McalM = pi(B)'' and mathcal M_icalM_i = pi(B_i)''. On the dense
+Proof. Let (H_phi, pi, Ω) be the tracial GNS representation and
+put M = pi(B)'' and M_i = pi(B_i)''. On the dense
subspace of classes [a], a in B, right multiplication
R_b[a] = [ab] is bounded with norm at most |b| by traciality, and it
commutes with pi(B) while the vectors R_b Ω = [b] are dense. If
-T in mathcal McalM has T Ω = 0, then T R_b Ω = R_b T Ω = 0
-for all b, so T = 0. Thus Ω is separating for mathcal M,calM, and
+T in M has T Ω = 0, then T R_b Ω = R_b T Ω = 0
+for all b, so T = 0. Thus Ω is separating for M, and
the normal vector state tau(T) = < T Ω, Ω > is
faithful: for T ge 0, tau(T) = 0 forces T^{1/2}Ω = 0. Its
-traciality passes from pi(B) to mathcal McalM by separate ultraweak
-continuity, so mathcal McalM is a finite von Neumann algebra with faithful
+traciality passes from pi(B) to M by separate ultraweak
+continuity, so M is a finite von Neumann algebra with faithful
normal trace tau.
-Extremality forces mathcal McalM to be a factor. A nontrivial central
-projection p in mathcal McalM would give the convex decomposition
+Extremality forces M to be a factor. A nontrivial central
+projection p in M would give the convex decomposition
- varphi(a) = tau(p) (tau(p pi(a)))/(tau(p)) + tau(1-p) (tau((1-p)pi(a)))/(tau(1-p)),
+ phi(a) = tau(p) (tau(p pi(a)))/(tau(p)) + tau(1-p) (tau((1-p)pi(a)))/(tau(1-p)),
and the two normalized summands are distinct tracial states on B: were the
-first equal to varphi, equality of the normal functionals would extend
-from pi(B) to mathcal McalM and evaluation at p would give
+first equal to phi, equality of the normal functionals would extend
+from pi(B) to M and evaluation at p would give
1 = tau(p), contradicting 0 < tau(p) < 1.
-The algebras mathcal M_icalM_i commute and generate mathcal M,calM, so each
-center satisfies Z(mathcal M_i)calM_i) ⊆ Z(mathcal M)calM) = mathbb CbbC 1,
-and tau makes every mathcal M_icalM_i a finite factor with a faithful normal
+The algebras M_i commute and generate M, so each
+center satisfies Z(M_i) ⊆ Z(M) = C 1,
+and tau makes every M_i a finite factor with a faithful normal
tracial state. Fix i and let y ge 0 lie in the von Neumann algebra
-generated by the other mathcal M_j.calM_j. It commutes with mathcal M_i,calM_i, and
+generated by the other M_j. It commutes with M_i, and
the functional
- psi_y(x) = tau(y^{1/2} x y^{1/2}) = tau(x y), x in mathcal M_i,calM_i,
+ psi_y(x) = tau(y^{1/2} x y^{1/2}) = tau(x y), x in M_i,
is normal, positive, tracial, and bounded by |y| tau, since
sandwiching by y^{1/2} preserves suprema of bounded increasing nets. The
Radon-Nikodym theorem for normal functionals on a finite von Neumann
-algebra gives h in mathcal M_icalM_i with 0 le h le |y| 1 and
+algebra gives h in M_i with 0 le h le |y| 1 and
psi_y(x) = tau(hx). Invariance of psi_y under unitary conjugation
-forces h to commute with every unitary of mathcal M_i,calM_i, hence to be
+forces h to commute with every unitary of M_i, hence to be
central and scalar; evaluation at the unit identifies the scalar as
tau(y). Thus tau(x y) = tau(x) tau(y). Linearity extends this to
all y in the algebra generated by the other rows, and applying it
@@ -243,27 +243,27 @@ The joint limit of these variables is Gaussian, and its covariance is the
limit of the empirical covariances.
Proposition. Let J ⊆ G be nonempty and finite, let
-N_j to ∞, and let varphi_j be extreme points of
-mathcal T(C)calT(C) with varphi_j(k_{N_j;g,h}) to R_{g,h} for
+N_j to ∞, and let phi_j be extreme points of
+T(C) with phi_j(k_{N_j;g,h}) to R_{g,h} for
g, h in J. Then R = (R_{g,h}) is a real positive semidefinite matrix
with diagonal entries one. Choose unit vectors v_g with
< v_g, v_h > = R_{g,h}, spanning a finite-dimensional real
Euclidean space L, let W be standard Gaussian in L, and put
xi_g = < v_g, W >. Then the pairs
((z_{N_j,g})_g, (k_{N_j;g,h})_{g,h}) converge in all ordered joint
-moments under varphi_j to ((xi_g)_g, (R_{g,h})_{g,h}). Equivalently,
+moments under phi_j to ((xi_g)_g, (R_{g,h})_{g,h}). Equivalently,
for every noncommutative *-polynomial P in these self-adjoint variables,
- varphi_j(P((z_{N_j,g})_g, (k_{N_j;g,h})_{g,h})) → mathbb EbbE, P((xi_g)_g, (R_{g,h})_{g,h}).
+ phi_j(P((z_{N_j,g})_g, (k_{N_j;g,h})_{g,h})) → E, P((xi_g)_g, (R_{g,h})_{g,h}).
-For every sequence (N_j, varphi_j) a subsequence satisfies the covariance
+For every sequence (N_j, phi_j) a subsequence satisfies the covariance
convergence.
-Proof. For a tracial varphi and self-adjoint x, y one has
-varphi(xy) = varphi(yx) = varphi(xy)bar, so the second moments
+Proof. For a tracial phi and self-adjoint x, y one has
+phi(xy) = phi(yx) = phi(xy)bar, so the second moments
are real and symmetric. For real scalars a_g,
- sum_{g,h in J} a_g a_h varphi_j(k_{N_j;g,h}) = (1)/(N_j) sum_{i=1}^{N_j} varphi_j((sum_{g in J} a_g x_{i,g})^2) ge 0.
+ sum_{g,h in J} a_g a_h phi_j(k_{N_j;g,h}) = (1)/(N_j) sum_{i=1}^{N_j} phi_j((sum_{g in J} a_g x_{i,g})^2) ge 0.
The diagonal entries are one by the row normalization, and Cauchy-Schwarz
bounds every entry by one in absolute value. The covariance matrices
@@ -272,10 +272,10 @@ have the asserted properties, and the subsequence assertion follows. A Gram
representation of R supplies L and the vectors v_g.
Consider a word of length d ge 1 in the fluctuation variables. At one
-term of the sequence write N = N_j, varphi = varphi_j. Expanding the
+term of the sequence write N = N_j, phi = phi_j. Expanding the
sums gives
- varphi(z_{N,g_1} ... z_{N,g_d}) = N^{-d/2} sum_{i_1, ..., i_d = 1}^N varphi(x_{i_1,g_1} ... x_{i_d,g_d}).
+ phi(z_{N,g_1} ... z_{N,g_d}) = N^{-d/2} sum_{i_1, ..., i_d = 1}^N phi(x_{i_1,g_1} ... x_{i_d,g_d}).
Partition the positions by equality of row labels. Distinct rows commute,
so the factors belonging to one row can be grouped while preserving their
@@ -284,56 +284,56 @@ rows. A singleton block gives zero by the row normalization, and each
surviving summand has absolute value at most M^d. A partition with b
blocks has at most N^b choices of distinct row labels, so its
contribution is at most M^d N^{b - d/2}. Among partitions without
-singleton blocks, b le lfloor d/2 rfloor. Hence odd moments tend to
+singleton blocks, b le floor( d/2 ). Hence odd moments tend to
zero, and for d = 2q only pairings into q pairs can survive.
Fix such a pairing with pairs (a_l, b_l), 1 le l le q. Its
contribution is
- N^{-q} sum_{substack{1 le i_1, ..., i_q le N i_1, ..., i_q distinct}} prod_{l=1}^q varphi(x_{i_l,g_{a_l}} x_{i_l,g_{b_l}}).
+ N^{-q} sum_{substack{1 le i_1, ..., i_q le N i_1, ..., i_q distinct}} prod_{l=1}^q phi(x_{i_l,g_{a_l}} x_{i_l,g_{b_l}}).
Removing the distinctness condition changes this by O_{d,M}(N^{-1}): there
are at most binom{q}{2} N^{q-1} tuples with a collision, and each product
is bounded by M^{2q}. The unrestricted sum factors, and traciality
identifies the factors with the empirical covariances:
- prod_{l=1}^q (frac1N sum_{i=1}^N varphi(x_{i,g_{a_l}} x_{i,g_{b_l}})) = prod_{l=1}^q varphi(k_{N;g_{a_l},g_{b_l}}).
+ prod_{l=1}^q (frac1N sum_{i=1}^N phi(x_{i,g_{a_l}} x_{i,g_{b_l}})) = prod_{l=1}^q phi(k_{N;g_{a_l},g_{b_l}}).
The error estimates depend only on d and M, so these are uniform in the
-moving traces varphi_j. By the covariance convergence the limiting even
+moving traces phi_j. By the covariance convergence the limiting even
moment is the sum over pairings of the corresponding products of entries of
R. This is exactly the Gaussian moment: the moment generating function
- mathbb EbbE exp(sum_{g in J} t_g xi_g) = exp(frac12 sum_{g,h in J} t_g t_h R_{g,h})
+ E exp(sum_{g in J} t_g xi_g) = exp(frac12 sum_{g,h in J} t_g t_h R_{g,h})
has the same Taylor expansion.
It remains to place the covariance variables in arbitrary word positions.
For g, h in J define the row covariances and the row-centered deviations
- s_{i;g,h} = frac{x_{i,g} x_{i,h} + x_{i,h} x_{i,g}}{2}, y_{j,i;g,h} = s_{i;g,h} - varphi_j(s_{i;g,h}) 1.
+ s_{i;g,h} = frac{x_{i,g} x_{i,h} + x_{i,h} x_{i,g}}{2}, y_{j,i;g,h} = s_{i;g,h} - phi_j(s_{i;g,h}) 1.
The centering is row by row at the current trace, so
-varphi_j(y_{j,i;g,h}) = 0 and |y_{j,i;g,h}| le 2M^2, and there is
+phi_j(y_{j,i;g,h}) = 0 and |y_{j,i;g,h}| le 2M^2, and there is
the identity
- k_{N_j;g,h} - varphi_j(k_{N_j;g,h}) 1 = (1)/(N_j) sum_{i=1}^{N_j} y_{j,i;g,h}.
+ k_{N_j;g,h} - phi_j(k_{N_j;g,h}) 1 = (1)/(N_j) sum_{i=1}^{N_j} y_{j,i;g,h}.
Let U_j be a word having d fluctuation factors and r ge 1 centered
covariance factors of the last form, expanded over rows. Each of its d + r
-row factors is centered under varphi_j, and a singleton row again
+row factors is centered under phi_j, and a singleton row again
vanishes by the independence lemma, so a surviving term uses at most
-lfloor (d+r)/2 rfloor distinct row labels. Absolute expectations within
+floor( (d+r)/2 ) distinct row labels. Absolute expectations within
one row are bounded by the product of the operator norms of the factors.
Finitely many partitions of the d+r positions occur, and with the
normalization N_j^{-d/2 - r} these observations give
- |varphi_j(U_j)| le C_{d,r,M} N_j^{lfloor (d+r)/2 rfloor - d/2 - r} le C_{d,r,M} N_j^{-r/2} → 0,
+ |phi_j(U_j)| le C_{d,r,M} N_j^{floor( (d+r)/2 ) - d/2 - r} le C_{d,r,M} N_j^{-r/2} → 0,
-with a constant independent of j and varphi_j.
+with a constant independent of j and phi_j.
Finally replace each covariance factor in a general word by its scalar
-varphi_j-mean plus the centered factor. Every term containing a centered
+phi_j-mean plus the centered factor. Every term containing a centered
covariance factor tends to zero by the last estimate, and the remaining
terms converge by the fluctuation computation and the covariance
convergence. This proves convergence of every ordered word. Since all
@@ -342,7 +342,7 @@ generators are self-adjoint, linearity gives the *-polynomial formulation.
The next step converts the Gaussian limit into noncommutative unitaries
that are almost invariant over the whole trace space. Retain the algebra
C, its invariant commuting orbit algebras C_i, and the variables
-x_{i,g}, z_{N,g}, k_{N;g,h} built above. Thus mathcal T(C)calT(C) is
+x_{i,g}, z_{N,g}, k_{N;g,h} built above. Thus T(C) is
nonempty, |x_{i,g}| le M, and the joint moment limit applies to any
sequence of extreme traces. Traces of finite-dimensional matrices in this
part are unnormalized, and for a matrix B write
@@ -353,9 +353,9 @@ Proposition. For every finite H subset G, every eps > 0, and every
integer m ge 1 there are t > 0 and a self-adjoint Q in C such that
V = exp(i t Q) satisfies
- sup_{varphi in mathcal T(C)}calT(C) |baralpha_h(V) - V|_{2,varphi} < eps (h in H),
+ sup_{phi in T(C)} |baralpha_h(V) - V|_{2,phi} < eps (h in H),
- sup_{varphi in mathcal T(C)}calT(C) |varphi(V^ell)| < eps (1 le ell le m).
+ sup_{phi in T(C)} |phi(V^ell)| < eps (1 le ell le m).
Three estimates are needed first.
@@ -378,7 +378,7 @@ Lemma. Let W be a standard Gaussian vector in a finite-dimensional real
Euclidean space and let B be a real symmetric operator there. Put
q_B = < W, BW > - Tr(B). Then
- |q_B|_{L^2} = √(2) |B|_{HS}, |mathbb EbbE exp(i s q_B)| le (1 + 4 s^2 |B|_{HS}^2)^{-1/4}
+ |q_B|_{L^2} = √(2) |B|_{HS}, |E exp(i s q_B)| le (1 + 4 s^2 |B|_{HS}^2)^{-1/4}
for every real s. The law of q_B has an exponential moment and is
determined by its moments.
@@ -388,7 +388,7 @@ q_B overset{law}{=} sum_j lambda_j(U_j^2 - 1) with independent
standard normals U_j. The summands are centered with variances
2lambda_j^2, giving the first formula. The one-dimensional Gaussian
integral gives
-|mathbb EbbE exp(i s q_B)| = prod_j (1 + 4 s^2 lambda_j^2)^{-1/4},
+|E exp(i s q_B)| = prod_j (1 + 4 s^2 lambda_j^2)^{-1/4},
and prod_j (1 + a_j) ge 1 + sum_j a_j for nonnegative a_j gives the
second formula.
@@ -397,11 +397,11 @@ The pointwise bound
exp(c|sum_j lambda_j(U_j^2 - 1)|) le exp(csum_j |lambda_j|) prod_j exp(c|lambda_j| U_j^2)
-and the Gaussian integral show mathbb EbbE exp(c|q_B|) < ∞. Let nu
+and the Gaussian integral show E exp(c|q_B|) < ∞. Let nu
be another probability measure with the same moments. By the nonnegative
power series for cosh and monotone convergence,
- int_{mathbb R}bbR cosh(cx), dnu(x) = sum_{r=0}^{∞} frac{c^{2r}}{(2r)!} mathbb E(q_B^bbE(q_B^{2r}) = mathbb EbbE cosh(c q_B) < ∞.
+ int_{R} cosh(cx), dnu(x) = sum_{r=0}^{∞} frac{c^{2r}}{(2r)!} E(q_B^{2r}) = E cosh(c q_B) < ∞.
Since e^{c|x|} le 2cosh(cx), the measure nu has an exponential
moment. The characteristic functions of nu and of q_B extend
@@ -420,13 +420,13 @@ variance at 2, and the perturbation lemma controls its change under
translation. The construction replaces the exact square root by a
polynomial with an explicitly controlled error.
-Lemma. Let X_j = X_j^* lie in unital C*-algebras with states varphi_j,
+Lemma. Let X_j = X_j^* lie in unital C*-algebras with states phi_j,
and let q_B be a centered Gaussian quadratic. If
-varphi_j(X_j^d) to mathbb E(q_B^d)bbE(q_B^d) for every d ge 0, then
-varphi_j(f(X_j)) to mathbb EbbE f(q_B) for every bounded continuous
-f : mathbb RbbR to mathbb C.bbC. No uniform bound on |X_j| is needed.
+phi_j(X_j^d) to E(q_B^d) for every d ge 0, then
+phi_j(f(X_j)) to E f(q_B) for every bounded continuous
+f : R to C. No uniform bound on |X_j| is needed.
-Proof. Let mu_j be the spectral law of X_j under varphi_j. The
+Proof. Let mu_j be the spectral law of X_j under phi_j. The
second moments are bounded, so the sequence is tight. Fix d ge 0 and an
even integer 2r > d. Moment convergence gives C_r < ∞ with
int |x|^{2r} dmu_j(x) le C_r for all j, hence
@@ -438,7 +438,7 @@ semicontinuity gives int |x|^{2r} dmu(x) le C_r, so the same tail bound
holds for mu. Multiply x^d by a continuous cutoff equal to one on
[-K, K] and zero outside [-2K, 2K]; weak convergence applies to this
bounded continuous function, and letting K to ∞ with the tail bound
-gives int x^d dmu(x) = mathbb E(q_B^d)bbE(q_B^d) for every d. The previous
+gives int x^d dmu(x) = E(q_B^d) for every d. The previous
lemma identifies mu with the law of q_B. Tightness then forces the
whole sequence mu_j to converge weakly to that law, since every
subsequence has a further weakly convergent subsequence with that same
@@ -464,7 +464,7 @@ vectors v_g with Gram matrix R in a finite-dimensional real Euclidean
space L, let W be standard Gaussian in L, and write
xi_g = < v_g, W >. For E = F or E = hF set
- P_E = frac1n sum_{g in E} v_g v_g^{mathsf T}sfT, B_E = p(P_E), q_E = < W, B_E W > - Tr(B_E).
+ P_E = frac1n sum_{g in E} v_g v_g^{T}, B_E = p(P_E), q_E = < W, B_E W > - Tr(B_E).
The operator P_E is positive with trace one and spectrum in [0,1], of
rank at most n. Because p(0) = 0, the operator p(P_E) - P_E^{1/2}
@@ -475,7 +475,7 @@ vanishes on the kernel of P_E, and the scalar approximation gives
hence 1 - eta le |B_E|_{HS} le 1 + eta.
The summands indexed by F cap hF cancel in P_F - P_{hF}, and each
-remaining v_g v_g^{mathsf T}sfT has trace norm one, so the square-root
+remaining v_g v_g^{T} has trace norm one, so the square-root
lemma gives
|P_F^{1/2} - P_{hF}^{1/2}|_{HS}^2 le |P_F - P_{hF}|_1 le (|F ,△, hF|)/(n) < delta.
@@ -487,14 +487,14 @@ centered quadratic of B_{hF} - B_F, and the Gaussian quadratic lemma
gives |q_{hF} - q_F|_{L^2} le sqrt2(sqrtdelta + 2eta). Also, by
the nondegeneracy and eta < 1/2, for every integer ell ge 1,
- |mathbb EbbE exp(iell t q_F)| le (1 + 4ell^2 t^2 |B_F|_{HS}^2)^{-1/4} le (1 + t^2)^{-1/4}.
+ |E exp(iell t q_F)| le (1 + 4ell^2 t^2 |B_F|_{HS}^2)^{-1/4} le (1 + t^2)^{-1/4}.
All these bounds hold for every such R, singular or not, and no
translation invariance of R has been used.
Step 3, the equivariant polynomial prescription. Direct multiplication of
the rank-one summands gives
-P_E^r = n^{-r} sum_{g_1, ..., g_r in E} R_{g_1,g_2} ... R_{g_{r-1},g_r} v_{g_1} v_{g_r}^{mathsf T}sfT.
+P_E^r = n^{-r} sum_{g_1, ..., g_r in E} R_{g_1,g_2} ... R_{g_{r-1},g_r} v_{g_1} v_{g_r}^{T}
Taking quadratic forms and traces shows
q_E = sum_{r=1}^D (c_r)/(n^r) sum_{g_1, ..., g_r in E} R_{g_1,g_2} ... R_{g_{r-1},g_r} (xi_{g_1} xi_{g_r} - R_{g_r,g_1}),
@@ -518,17 +518,17 @@ commutes with baralpha_h. Hence
Q_{N,hF} = baralpha_h(Q_{N,F}) (h in H, N ge 1).
Step 4, transfer of moments and scalar laws. Take any N_j to ∞ and
-extreme traces varphi_j in mathcal T(C)calT(C) with
-varphi_j(k_{N_j;g,g'}) to R_{g,g'} for g, g' in J. The joint moment
+extreme traces phi_j in T(C) with
+phi_j(k_{N_j;g,g'}) to R_{g,g'} for g, g' in J. The joint moment
limit gives convergence of every ordered mixed moment of the empirical
variables, and any moment of the polynomials just written is a finite
linear combination of these. Since the limiting variables commute and the
closed formula for q_E is real-valued, the Q_{N_j,E} converge jointly
-in moments under varphi_j to the q_E. In particular
+in moments under phi_j to the q_E. In particular
- |Q_{N_j,hF} - Q_{N_j,F}|_{2,varphi_j} → |q_{hF} - q_F|_{L^2},
+ |Q_{N_j,hF} - Q_{N_j,F}|_{2,phi_j} → |q_{hF} - q_F|_{L^2},
- varphi_j(exp(i s Q_{N_j,F})) → mathbb EbbE exp(i s q_F)
+ phi_j(exp(i s Q_{N_j,F})) → E exp(i s q_F)
for every real s. The passage to scalar spectral laws is needed because
the operator norms of Q_{N,E} may grow with N: the estimates
@@ -538,43 +538,43 @@ together with the spectral tail estimate is enough for the exponential
convergence.
Step 5, uniformity over all traces. Set V_N = exp(i t Q_{N,F}).
-For self-adjoint X, Y in a unital C*-algebra and a tracial varphi,
+For self-adjoint X, Y in a unital C*-algebra and a tracial phi,
differentiation gives
(d)/(ds)(e^{i(1-s)tX} e^{istY}) = e^{i(1-s)tX}, it(Y - X), e^{istY},
and the tracial 2-seminorm is unchanged by left and right unitary
multiplication, so integration and the triangle inequality give the
-tracial Duhamel bound |e^{itX} - e^{itY}|_{2,varphi} le t|X - Y|_{2,varphi}.
+tracial Duhamel bound |e^{itX} - e^{itY}|_{2,phi} le t|X - Y|_{2,phi}.
Together with exact equivariance, the norm limit, and the Gaussian
translation bound, every sequence of traces and indices with convergent
covariance satisfies
- limsup_{j to ∞} |baralpha_h(V_{N_j}) - V_{N_j}|_{2,varphi_j} le tsqrt2(sqrtdelta + 2eta) < eps/4,
+ limsup_{j to ∞} |baralpha_h(V_{N_j}) - V_{N_j}|_{2,phi_j} le tsqrt2(sqrtdelta + 2eta) < eps/4,
- lim_{j to ∞} |varphi_j(V_{N_j}^ell)| = |mathbb EbbE exp(iell t q_F)| le (1 + t^2)^{-1/4} < eps/4
+ lim_{j to ∞} |phi_j(V_{N_j}^ell)| = |E exp(iell t q_F)| le (1 + t^2)^{-1/4} < eps/4
for h in H and 1 le ell le m, by the choices of the parameters.
Consequently, for all sufficiently large N, every extreme trace
-varphi in mathcal T(C)calT(C) satisfies
-|baralpha_h(V_N) - V_N|_{2,varphi} < eps/2 and
-|varphi(V_N^ell)| < eps/2 for h in H and 1 le ell le m. Were
-this false, some N_j to ∞ and extreme traces varphi_j would fail
+phi in T(C) satisfies
+|baralpha_h(V_N) - V_N|_{2,phi} < eps/2 and
+|phi(V_N^ell)| < eps/2 for h in H and 1 le ell le m. Were
+this false, some N_j to ∞ and extreme traces phi_j would fail
one of these finitely many tests at tolerance eps/2. The covariance
-matrices (varphi_j(k_{N_j;g,g'}))_{g,g' in J} lie in the compact set of
+matrices (phi_j(k_{N_j;g,g'}))_{g,g' in J} lie in the compact set of
real positive semidefinite matrices with diagonal one, so a subsequence has
a limiting covariance R, and the sequential estimates contradict the
failure. Only the finite covariance matrices were extracted; no compactness
of the extreme trace boundary is used.
Fix N with these extreme-trace estimates. For each h in H the squared
-seminorm varphi mapsto varphi((baralpha_h(V_N) - V_N)^*(baralpha_h(V_N) - V_N))
-is weak-star continuous and affine on the convex set mathcal T(C),calT(C), and
-for each 1 le ell le m the map varphi mapsto |varphi(V_N^ell)|
+seminorm phi |-> phi((baralpha_h(V_N) - V_N)^*(baralpha_h(V_N) - V_N))
+is weak-star continuous and affine on the convex set T(C), and
+for each 1 le ell le m the map phi |-> |phi(V_N^ell)|
is weak-star continuous and convex. Their sublevel sets with bounds
eps^2/4 and eps/2 are closed and convex and contain every extreme
-trace, so by the Krein-Milman theorem these sublevel sets contain mathcal T(C).calT(C). Hence
+trace, so by the Krein-Milman theorem these sublevel sets contain T(C). Hence
both suprema in the proposition are at most eps/2 < eps, and
Q = Q_{N,F}, V = V_N complete the proof.
@@ -585,16 +585,16 @@ coordinate-selection fact.
Lemma. If x = [(x_j)] in A^omega has a bounded representative, then
- sup_{tau in mathcal T_calT_omega(A)} |x|_{2,tau} = lim_{j to omega} sup_{sigma in mathcal T(A)}calT(A) |x_j|_{2,sigma},
+ sup_{tau in T_} |x|_{2,tau} = lim_{j to omega} sup_{sigma in T(A)} |x_j|_{2,sigma},
- sup_{tau in mathcal T_calT_omega(A)} |tau(x)| = lim_{j to omega} sup_{sigma in mathcal T(A)}calT(A) |sigma(x_j)|.
+ sup_{tau in T_} |tau(x)| = lim_{j to omega} sup_{sigma in T(A)} |sigma(x_j)|.
Finitely many such tests can therefore be realized simultaneously at a
single coordinate with any prescribed positive slack above their
ultralimit bounds.
Proof. Every sequence of traces gives the upper bound in each identity.
-Conversely, for each j choose sigma_j in mathcal T(A)calT(A) within 1/j of
+Conversely, for each j choose sigma_j in T(A) within 1/j of
the supremum for the test in question. Its limit trace realizes the reverse
inequality, since square roots and absolute values commute with ultralimits
of bounded scalar sequences. For finitely many tests, the coordinates
@@ -605,7 +605,7 @@ differ from test to test.
Keep the invariant separable algebra C subset A^omega cap A'
constructed from the rows. For a in A_+ with 0 le a le 1 and
-tau in mathcal T_calT_omega(A), define on C
+tau in T_, define on C
tau_a(c) = tau(ac).
@@ -613,11 +613,11 @@ This is a positive tracial functional: positivity follows from
ac = a^{1/2} c a^{1/2} when c ge 0, using that a commutes with C,
and traciality from tau(acd) = tau(dac) = tau(adc). Its norm is
tau(a). When tau(a) > 0 its normalization is an element of
-mathcal T(C),calT(C), and when tau(a) = 0 it vanishes, so the uniform
+T(C), and when tau(a) = 0 it vanishes, so the uniform
approximately invariant unitary V = exp(i t Q) from the
preceding proposition gives
- |tau(a V^ell)| < eps, tau(a) le eps (0 le a le 1, tau in mathcal T_calT_omega(A), 1 le ell le m, tau(a) > 0),
+ |tau(a V^ell)| < eps, tau(a) le eps (0 le a le 1, tau in T_, 1 le ell le m, tau(a) > 0),
with zero left side when tau(a) = 0. Approximate invariance passes to
limit traces by restriction to C.
@@ -625,10 +625,10 @@ limit traces by restriction to C.
Proposition. There is a unitary w in (A^omega cap A')^{baralpha}
with
- tau(a f(w)) = tau(a) int_{mathbb T}bbT f, dmu
+ tau(a f(w)) = tau(a) int_{T} f, dmu
-for every a in A, every tau in mathcal T_calT_omega(A), and every
-continuous f, where mathbb TbbT = {z in mathbb CbbC : |z| = 1} and mu
+for every a in A, every tau in T_, and every
+continuous f, where T = {z in C : |z| = 1} and mu
is normalized Haar measure.
Proof. Choose increasing finite sets H_m subset G with union G and a
@@ -648,7 +648,7 @@ j(m) ge m at which w_m = v^{(m)}_{j(m)} satisfies
|alpha_h(w_m) - w_m|_{2,u} < 2/m (h in H_m),
- sup_{sigma in mathcal T(A)}calT(A) |sigma(a_r w_m^ell)| < 2/m (1 le r, ell le m).
+ sup_{sigma in T(A)} |sigma(a_r w_m^ell)| < 2/m (1 le r, ell le m).
The last display applies the trace identity of the lemma to the class
a_r (V^{(m)})^ell. All coordinate tests are finite in number, and their
@@ -658,21 +658,21 @@ The class w = [(w_m)] is a unitary. The first two displays, density, and
|[w_m, a - a_r]|_{2,u} le 2|a - a_r| make it central relative to A
and fixed by baralpha. The last display, first for every a_r, then by
norm density for positive contractions, then by linearity for a in A,
-gives tau(a w^ell) = 0 for every a in A, tau in mathcal T_calT_omega(A),
+gives tau(a w^ell) = 0 for every a in A, tau in T_,
and ell ge 1. The same holds for negative powers because
tau(a w^{-ell}) = tau(a^* w^ell)bar, and the zeroth power has
trace tau(a). Hence the identity holds for trigonometric polynomials,
-and uniform approximation on mathbb TbbT together with
+and uniform approximation on T together with
|tau(a f(w))| le |a| |f|_∞ proves it for every continuous f.
Proposition. The action alpha has equivariant uniform property Gamma,
and its projections can be chosen to sum to 1.
Proof. Fix k ge 2 and let w = [(w_j)] be the unitary just constructed,
-represented by unitaries w_j in A. Partition mathbb TbbT into k
+represented by unitaries w_j in A. Partition T into k
half-open arcs I_1, ..., I_k of Haar measure 1/k and write
chi_i = 1_{I_i}. Choose continuous functions
-f_i^{(r)} : mathbb TbbT to [0,1] with
+f_i^{(r)} : T to [0,1] with
e_r := max_{1 le i le k} |f_i^{(r)} - chi_i|_{L^2(mu)} → 0,
@@ -682,7 +682,7 @@ e = e_r. Pointwise bounds and the triangle inequality give
|f_i^2 - f_i|_{L^2(mu)} le e, |f_i f_j|_{L^2(mu)} le 2e (i ne j),
- Bigl|sum_i f_i - 1Bigr|_{L^2(mu)} le k e, |int_{mathbb T}bbT f_i, dmu - frac1k| le e.
+ Bigl|sum_i f_i - 1Bigr|_{L^2(mu)} le k e, |int_{T} f_i, dmu - frac1k| le e.
The first uses |f_i^2 - f_i| le |f_i - chi_i| almost everywhere; the
second expands
@@ -695,10 +695,10 @@ Set y_i^{(r)} = f_i^{(r)}(w). These are central, invariant positive
contractions because continuous functional calculus respects commutation and
automorphisms. The trace identity with a = 1, applied to the squared
moduli of the functions above, gives the corresponding 2-norm bounds for
-the y_i^{(r)} uniformly over tau in mathcal T_calT_omega(A), and for
+the y_i^{(r)} uniformly over tau in T_, and for
every positive contraction a in A it also gives
- |tau(a y_i^{(r)}) - (tau(a))/(k)| = tau(a) |int_{mathbb T}bbT f_i^{(r)}, dmu - frac1k| le e_r.
+ |tau(a y_i^{(r)}) - (tau(a))/(k)| = tau(a) |int_{T} f_i^{(r)}, dmu - frac1k| le e_r.
Represent y_i^{(r)} by f_i^{(r)}(w_j), and use H_r and
a_1, ..., a_r from the preceding proof. For fixed r the
@@ -712,7 +712,7 @@ indicated indices,
|[y_{i,r}, a_s]|_{2,u} < 1/r (s le r), |alpha_h(y_{i,r}) - y_{i,r}|_{2,u} < 1/r (h in H_r),
- sup_{sigma in mathcal T(A)}calT(A) |sigma(a_s y_{i,r}) - (sigma(a_s))/(k)| < e_r + 1/r (s le r).
+ sup_{sigma in T(A)} |sigma(a_s y_{i,r}) - (sigma(a_s))/(k)| < e_r + 1/r (s le r).
This is a coordinate choice made after f_i^{(r)} is fixed, so it needs no
uniform continuity modulus, as r varies, for these increasingly sharp arc
@@ -722,7 +722,7 @@ Let p_i = [(y_{i,r})_r] in A^omega. The representatives are uniformly
bounded positive contractions. Small square defect, orthogonality, and unit
sum make the p_i orthogonal projections summing to 1. Small commutators
and approximate invariance make them central and invariant by density. The
-trace estimate gives, for every sequence (sigma_r) in mathcal T(A),calT(A),
+trace estimate gives, for every sequence (sigma_r) in T(A),
lim_{r to omega} sigma_r(a_s y_{i,r}) = frac1k lim_{r to omega} sigma_r(a_s),