diff b1f9bc4..56bc728 equivariant_jiang_su_stability_v2.txt
diff --git a/equivariant_jiang_su_stability_v2.txt b/equivariant_jiang_su_stability_v2.txt
index e6eceda..ef854c0 100644
--- a/equivariant_jiang_su_stability_v2.txt
+++ b/equivariant_jiang_su_stability_v2.txt
@@ -1,41 +1,41 @@
-A C*-algebra is Jiang-Su stable when it absorbs the Jiang-Su algebra $Z$
+A C*-algebra is Jiang-Su stable when it absorbs the Jiang-Su algebra Z
tensorially. The dynamic counterpart asks an action to absorb the trivial
-action on $Z$ up to cocycle conjugacy. The main theorem establishes this
+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
-$B$, and $Z$ is the Jiang-Su algebra.
+minimal, mathcal T(B)calT(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
+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 cong 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
-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
-$u_g in B$ for $g in G$ such that
+Theorem. Let A be a simple, separable, unital, infinite-dimensional,
+nuclear, stably finite C*-algebra with A cong 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
+u_g in B for g in G such that
-$$ u_e = 1, qquad u_{gh} = u_g beta_g(u_h), qquad Phi(alpha_g(a)) = u_g beta_g(Phi(a)) u_g^* $$
+ u_e = 1, u_{gh} = u_g beta_g(u_h), Phi(alpha_g(a)) = u_g beta_g(Phi(a)) u_g^*
-for all $g, h in G$ and $a in A$.
+for all g, h in G and a in A.
Stable finiteness means that every isometry in every matrix algebra over
-$A$ is a unitary. Amenability enters only through the Folner condition:
-for every finite $H subset G$ and every $delta > 0$ there is a nonempty
-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
-$g mapsto u_g$.
+A is a unitary. Amenability enters only through the Folner condition:
+for every finite H subset G and every delta > 0 there is a nonempty
+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.
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
+and no condition is placed on the action of G on the tracial boundary or
on individual trace orbits. Finite groups and the trivial group are
included. The statement is the unital, stably finite case of Szabo's
conjecture that amenable actions on separable simple nuclear Jiang-Su
@@ -53,18 +53,18 @@ comparison of positive elements.
./pdf/matui-sato-1111.1637-strict-comparison-Z-absorption.pdf
-Theorem 1.1 (Matui and Sato). Let $A$ be a unital separable simple
+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: $A ⊗ Z cong A$; $A$
-has strict comparison; every completely positive map from $A$ to $A$
+traces. Then the following are equivalent: A ⊗ Z cong A; A
+has strict comparison; every completely positive map from A to A
can be excised in small central sequences; has property (SI).
The decisive criterion reads:
-Corollary C (Szabo and Wouters). Let $alpha : G curvearrowright A$ be
+Corollary C (Szabo and Wouters). Let alpha : G curvearrowright A be
an action of a countable amenable group on a separable unital simple
-nuclear $Z$-stable C*-algebra. Then $alpha$ has equivariant uniform
-property Gamma if and only if $alpha$ is equivariantly Jiang-Su stable.
+nuclear Z-stable C*-algebra. Then alpha has equivariant uniform
+property Gamma if and only if alpha is equivariantly Jiang-Su stable.
./pdf/szabo-wouters-2301.12846-equivariant-property-gamma.pdf
@@ -72,13 +72,13 @@ 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
-$|x|_{2,tau} = tau(x^*x)^{1/2}$ and
-$|x|_{2,u} = sup_{tau in mathcal T(B)}calT(B) |x|_{2,tau}$. Under the
-hypotheses on $A$, nuclearity implies exactness, and a unital stably
+For a unital C*-algebra B and tau in mathcal T(B)calT(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
+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
-compact convex set, and each trace on the simple algebra $A$ is faithful,
+unital exact algebra is a trace. Hence mathcal T(A)calT(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.
@@ -87,253 +87,253 @@ unit.
Theorem 5.11 (Haagerup). Quasitraces on exact unital C*-algebras are traces.
Corollary 5.12 (Haagerup). Every stably finite unital exact C*-algebra has a trace state.
-Fix a free ultrafilter $omega$ on the positive integers and form the
+Fix a free ultrafilter omega on the positive integers and form the
uniform tracial ultrapower
-$$ A^omega = ell^∞(mathbb N,bbN, A) / I_omega, qquad I_omega = {(a_j) : lim_{j to omega} |a_j|_{2,u} = 0}. $$
-
-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
-$tau([(a_j)]) = lim_{j to omega} tau_j(a_j)$ 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
-applying $alpha_g$ coordinatewise gives an action
-$baralpha : G curvearrowright A^omega$ preserving $A^omega cap A'$.
-
-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
-forces $sum_i p_i = 1$. The absorption criterion to be used states that a
+ A^omega = ell^∞(mathbb N,bbN, 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
+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
+applying alpha_g coordinatewise gives an action
+baralpha : G curvearrowright A^omega preserving A^omega cap A'.
+
+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
+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
equivariant uniform property Gamma, is cocycle conjugate to its tensor
-product with the trivial action on $Z$. It therefore remains to produce
+product with the trivial action on Z. It therefore remains to produce
the projections, working inside a separable subalgebra of
-$A^omega cap A'$ and testing against all of its traces, not only those
+A^omega cap A' and testing against all of its traces, not only those
extending to the ambient algebra.
-The first construction plants many commuting copies of the orbit of $Z$ in
-the central sequence algebra. Write $C^*(S)$ for the unital C*-algebra
-generated by a set $S$.
+The first construction plants many commuting copies of the orbit of Z in
+the central sequence algebra. Write C^*(S) for the unital C*-algebra
+generated by a set S.
-Proposition. There are unital embeddings $iota_i : Z to A^omega$,
-$i ge 1$, whose orbit algebras
+Proposition. There are unital embeddings iota_i : Z to A^omega,
+i ge 1, whose orbit algebras
-$$ C_i = C^*(baralpha_g(iota_i(Z)) : g in G) $$
+ C_i = C^*(baralpha_g(iota_i(Z)) : g in G)
-commute with $A$ and commute pairwise. Consequently
+commute with A and commute pairwise. Consequently
-$$ C = C^*(∪_{i ge 1} C_i) ⊆ A^omega cap A' $$
+ 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
-$x_{i,g} in C_i$ and a constant $M < ∞$ with
+is a unital separable baralpha-invariant subalgebra with
+mathcal T(C)calT(C) neq ∅. There are moreover self-adjoint elements
+x_{i,g} in C_i and a constant M < ∞ with
-$$ |x_{i,g}| le M, qquad varphi(x_{i,g}) = 0, qquad varphi(x_{i,g}^2) = 1 quad (varphi in mathcal T(C)),calT(C)), qquad baralpha_h(x_{i,g}) = x_{i,hg} $$
+ |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}
-for all $i ge 1$ and $g, h in G$.
+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
+Proof. Since A is Z-stable, fix an identification
+A cong 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
-representatives $d_s = [(d_{s,j})_{j ge 1}]$. For each $j$ choose a finite
-tensor stage containing elements within $1/j$ in norm of
-$d_{1,j}, ..., d_{j,j}$, and let $theta_j : Z to A$ embed into a tensor
-factor strictly after that stage. For $j ge s$ and $z in Z$,
+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
+representatives d_s = [(d_{s,j})_{j ge 1}]. For each j choose a finite
+tensor stage containing elements within 1/j in norm of
+d_{1,j}, ..., d_{j,j}, and let theta_j : Z to A embed into a tensor
+factor strictly after that stage. For j ge s and z in Z,
-$$ |[theta_j(z), d_{s,j}]| le (2|z|)/(j). $$
+ |[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 $Z$
-is simple and the unit of $A^omega$ is nonzero.
+z mapsto [(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
-$baralpha$-invariant subalgebra of $A^omega$ containing $A$ and the
+Set D_0 = A. Suppose D_{i-1} is a separable unital
+baralpha-invariant subalgebra of A^omega containing A and the
preceding orbit algebras. The embedding just proved places
-$iota_i(Z)$ in $D_{i-1}'$; define $C_i$ as above and
-$D_i = C^*(D_{i-1}, C_i)$. For $b in iota_i(Z)$, $d in D_{i-1}$, and
-$g in G$, invariance of $D_{i-1}$ gives
+iota_i(Z) in D_{i-1}'; define C_i as above and
+D_i = C^*(D_{i-1}, C_i). For b in iota_i(Z), d in D_{i-1}, and
+g in G, invariance of D_{i-1} gives
-$$ [baralpha_g(b), d] = baralpha_g([b, baralpha_{g^{-1}}(d)]) = 0, $$
+ [baralpha_g(b), d] = baralpha_g([b, baralpha_{g^{-1}}(d)]) = 0,
-so $C_i$ commutes with $D_{i-1}$. Countability of $G$ keeps each $D_i$
+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
-$mathcal T(C)calT(C) neq ∅$.
-
-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
-translated unital embedding of $Z$ to $rho$, giving the normalization, and
+A^omega restricts to a tracial state on C, so
+mathcal T(C)calT(C) neq ∅.
+
+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
+translated unital embedding of Z to rho, giving the normalization, and
the action law gives the equivariance. This proves the proposition.
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
-
-$$ varphi(b_1 ... b_s) = prod_{j=1}^s varphi(b_j) $$
-
-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
-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
-$T in mathcal M$calM$ 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
-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 $mathcal M$calM$ by separate ultraweak
-continuity, so $mathcal M$calM$ is a finite von Neumann algebra with faithful
-normal trace $tau$.
-
-Extremality forces $mathcal M$calM$ to be a factor. A nontrivial central
-projection $p in mathcal M$calM$ would give the convex decomposition
-
-$$ varphi(a) = tau(p) (tau(p pi(a)))/(tau(p)) + tau(1-p) (tau((1-p)pi(a)))/(tau(1-p)), $$
-
-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 $mathcal M$calM$ and evaluation at $p$ would give
-$1 = tau(p)$, contradicting $0 < tau(p) < 1$.
-
-The algebras $mathcal M_i$calM_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 $mathcal M_i$calM_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
+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
+
+ varphi(b_1 ... b_s) = prod_{j=1}^s varphi(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
+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
+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
+normal trace tau.
+
+Extremality forces mathcal McalM to be a factor. A nontrivial central
+projection p in mathcal McalM 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)),
+
+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
+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
+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
the functional
-$$ psi_y(x) = tau(y^{1/2} x y^{1/2}) = tau(x y), qquad x in mathcal M_i,calM_i, $$
+ psi_y(x) = tau(y^{1/2} x y^{1/2}) = tau(x y), x in mathcal M_i,calM_i,
-is normal, positive, tracial, and bounded by $|y| tau$, since
-sandwiching by $y^{1/2}$ preserves suprema of bounded increasing nets. The
+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 $h in mathcal M_i$calM_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$, hence to be
+algebra gives h in mathcal M_icalM_i with 0 le h le |y| 1 and
+psi_y(x) = tau(hx). Invariance of psi_y under unitary conjugation
+forces h to commute with every unitary of mathcal M_i,calM_i, hence to be
central and scalar; evaluation at the unit identifies the scalar as
-$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
+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
successively gives the product formula. This proves the lemma.
-For $N ge 1$ and $g, h in G$ set
+For N ge 1 and g, h in G set
-$$ z_{N,g} = (1)/(sqrt N) sum_{i=1}^N x_{i,g}, qquad k_{N;g,h} = (1)/(N) sum_{i=1}^N frac{x_{i,g} x_{i,h} + x_{i,h} x_{i,g}}{2}. $$
+ z_{N,g} = (1)/(sqrt N) sum_{i=1}^N x_{i,g}, k_{N;g,h} = (1)/(N) sum_{i=1}^N frac{x_{i,g} x_{i,h} + x_{i,h} x_{i,g}}{2}.
-The last two subscripts of $k_{N;g,h}$ are covariance indices. These
-elements are self-adjoint, $k_{N;g,h} = k_{N;h,g}$, and
+The last two subscripts of k_{N;g,h} are covariance indices. These
+elements are self-adjoint, k_{N;g,h} = k_{N;h,g}, and
-$$ baralpha_l(z_{N,g}) = z_{N,lg}, qquad baralpha_l(k_{N;g,h}) = k_{N;lg,lh} qquad (l, g, h in G). $$
+ baralpha_l(z_{N,g}) = z_{N,lg}, baralpha_l(k_{N;g,h}) = k_{N;lg,lh} (l, g, h in G).
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
-$g, h in J$. 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,
-for every noncommutative *-polynomial $P$ in these self-adjoint variables,
-
-$$ varphi_j(P((z_{N_j,g})_g, (k_{N_j;g,h})_{g,h})) → mathbb EbbE, P((xi_g)_g, (R_{g,h})_{g,h}). $$
-
-For every sequence $(N_j, varphi_j)$ a subsequence satisfies the covariance
+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
+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,
+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}).
+
+For every sequence (N_j, varphi_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
-are real and symmetric. For real scalars $a_g$,
+Proof. For a tracial varphi and self-adjoint x, y one has
+varphi(xy) = varphi(yx) = varphi(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 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.
The diagonal entries are one by the row normalization, and Cauchy-Schwarz
bounds every entry by one in absolute value. The covariance matrices
therefore lie in a compact set of real symmetric matrices, their limits
have the asserted properties, and the subsequence assertion follows. A Gram
-representation of $R$ supplies $L$ and the vectors $v_g$.
+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
+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
sums gives
-$$ varphi(z_{N,g_1} ... z_{N,g_d}) = N^{-d/2} sum_{i_1, ..., i_d = 1}^N varphi(x_{i_1,g_1} ... x_{i_d,g_d}). $$
+ 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}).
Partition the positions by equality of row labels. Distinct rows commute,
so the factors belonging to one row can be grouped while preserving their
order, and the independence lemma factors the expectation over the distinct
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
-zero, and for $d = 2q$ only pairings into $q$ pairs can survive.
+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
+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
+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 varphi(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
+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 varphi(x_{i,g_{a_l}} x_{i,g_{b_l}})) = prod_{l=1}^q varphi(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
+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
moment is the sum over pairings of the corresponding products of entries of
-$R$. This is exactly the Gaussian moment: the moment generating function
+R. This is exactly the Gaussian moment: the moment generating function
-$$ mathbb EbbE exp(sum_{g in J} t_g xi_g) = exp(frac12 sum_{g,h in J} t_g t_h R_{g,h}) $$
+ 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})
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
+For g, h in J define the row covariances and the row-centered deviations
-$$ s_{i;g,h} = frac{x_{i,g} x_{i,h} + x_{i,h} x_{i,g}}{2}, qquad y_{j,i;g,h} = s_{i;g,h} - varphi_j(s_{i;g,h}) 1. $$
+ 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} - varphi_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
+varphi_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} - varphi_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
+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
vanishes by the independence lemma, so a surviving term uses at most
-$lfloor (d+r)/2 rfloor$ distinct row labels. Absolute expectations within
+lfloor (d+r)/2 rfloor 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
+Finitely many partitions of the d+r positions occur, and with the
+normalization N_j^{-d/2 - r} these observations give
-$$ |varphi_j(U_j)| le C_{d,r,M} N_j^{lfloor (d+r)/2 rfloor - d/2 - r} le C_{d,r,M} N_j^{-r/2} → 0, $$
+ |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,
-with a constant independent of $j$ and $varphi_j$.
+with a constant independent of j and varphi_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
+varphi_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
@@ -341,414 +341,414 @@ 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
-nonempty, $|x_{i,g}| le M$, and the joint moment limit applies to any
+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
+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
-$|B|_{HS} = (Tr(B^*B))^{1/2}$ and
-$|B|_1 = Tr|B|$.
+part are unnormalized, and for a matrix B write
+|B|_{HS} = (Tr(B^*B))^{1/2} and
+|B|_1 = Tr|B|.
-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
+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
-$$ sup_{varphi in mathcal T(C)}calT(C) |baralpha_h(V) - V|_{2,varphi} < eps quad (h in H), $$
+ sup_{varphi in mathcal T(C)}calT(C) |baralpha_h(V) - V|_{2,varphi} < eps (h in H),
-$$ sup_{varphi in mathcal T(C)}calT(C) |varphi(V^ell)| < eps quad (1 le ell le m). $$
+ sup_{varphi in mathcal T(C)}calT(C) |varphi(V^ell)| < eps (1 le ell le m).
Three estimates are needed first.
-Lemma. If $S, T$ are positive matrices of the same size, then
-$|S^{1/2} - T^{1/2}|_{HS}^2 le |S - T|_1$.
+Lemma. If S, T are positive matrices of the same size, then
+|S^{1/2} - T^{1/2}|_{HS}^2 le |S - T|_1.
-Proof. Set $a = S^{1/2}$, $b = T^{1/2}$, $d = a - b$, and
-$u = sign(d)$ with $sign(0) = 0$. Since
-$a^2 - b^2 = ad + db$, cyclicity of the trace gives
-$Tr(u(a^2 - b^2)) = Tr(|d|(a + b))$. Diagonalizing $d$
-gives $a_{jj} + b_{jj} ge |a_{jj} - b_{jj}| = |d_j|$ by positivity, so
+Proof. Set a = S^{1/2}, b = T^{1/2}, d = a - b, and
+u = sign(d) with sign(0) = 0. Since
+a^2 - b^2 = ad + db, cyclicity of the trace gives
+Tr(u(a^2 - b^2)) = Tr(|d|(a + b)). Diagonalizing d
+gives a_{jj} + b_{jj} ge |a_{jj} - b_{jj}| = |d_j| by positivity, so
-$$ Tr(|d|(a+b)) = sum_j |d_j|(a_{jj}+b_{jj}) ge sum_j d_j^2 = Tr(d^2). $$
+ Tr(|d|(a+b)) = sum_j |d_j|(a_{jj}+b_{jj}) ge sum_j d_j^2 = Tr(d^2).
-The operator $u$ is a contraction, so trace-norm duality gives
-$Tr(u(a^2 - b^2)) le Tr|a^2 - b^2|$. The two
+The operator u is a contraction, so trace-norm duality gives
+Tr(u(a^2 - b^2)) le Tr|a^2 - b^2|. The two
inequalities prove the lemma.
-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
+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
-$$ |q_B|_{L^2} = √(2) |B|_{HS}, qquad |mathbb EbbE exp(i s q_B)| le (1 + 4 s^2 |B|_{HS}^2)^{-1/4} $$
+ |q_B|_{L^2} = √(2) |B|_{HS}, |mathbb EbbE 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
+for every real s. The law of q_B has an exponential moment and is
determined by its moments.
Proof. Orthogonal diagonalization writes
-$q_B overset{law}{=} sum_j lambda_j(U_j^2 - 1)$ with independent
-standard normals $U_j$. The summands are centered with variances
-$2lambda_j^2$, giving the first formula. The one-dimensional Gaussian
+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
-$|mathbb EbbE exp(i s q_B)| = prod_j (1 + 4 s^2 lambda_j^2)^{-1/4}$,
-and $prod_j (1 + a_j) ge 1 + sum_j a_j$ for nonnegative $a_j$ gives the
+|mathbb EbbE 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.
-Choose $c > 0$ with $2 c max_j |lambda_j| < 1$, any $c$ when $B = 0$.
+Choose c > 0 with 2 c max_j |lambda_j| < 1, any c when B = 0.
The pointwise bound
-$$ exp(c|sum_j lambda_j(U_j^2 - 1)|) le exp(csum_j |lambda_j|) prod_j exp(c|lambda_j| U_j^2) $$
+ 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 mathbb EbbE exp(c|q_B|) < ∞. Let nu
be another probability measure with the same moments. By the nonnegative
-power series for $cosh$ and monotone convergence,
+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_{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) < ∞.
-Since $e^{c|x|} le 2cosh(cx)$, the measure $nu$ has an exponential
-moment. The characteristic functions of $nu$ and of $q_B$ extend
-analytically to the strip $|Im z| < c$, their Taylor
+Since e^{c|x|} le 2cosh(cx), the measure nu has an exponential
+moment. The characteristic functions of nu and of q_B extend
+analytically to the strip |Im z| < c, their Taylor
coefficients at zero agree, and the identity theorem makes them equal.
-Uniqueness of characteristic functions gives $nu$ equal to the law of
-$q_B$. This proves the lemma.
+Uniqueness of characteristic functions gives nu equal to the law of
+q_B. This proves the lemma.
The square-root normalization is exactly what keeps the variance away from
-zero. For a positive real symmetric matrix $P$ with $Tr(P) = 1$
-the variance formulas give $Var(q_{P^{1/2}}) = 2$ and
-$Var(q_P) = 2Tr(P^2)$. The latter equals $2$ for a
-rank-one projection but $2/n$ for $P = n^{-1} I_n$, so using $P$ directly
+zero. For a positive real symmetric matrix P with Tr(P) = 1
+the variance formulas give Var(q_{P^{1/2}}) = 2 and
+Var(q_P) = 2Tr(P^2). The latter equals 2 for a
+rank-one projection but 2/n for P = n^{-1} I_n, so using P directly
would make the variance vanish as the rank grows. The square root fixes the
-variance at $2$, and the perturbation lemma controls its change under
+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 $varphi_j$,
-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.
-
-Proof. Let $mu_j$ be the spectral law of $X_j$ under $varphi_j$. 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
-
-$$ int_{|x| > K} |x|^d, dmu_j(x) le C_r K^{d-2r} qquad (K > 0). $$
-
-Take any weakly convergent subsequence with limit $mu$. Lower
-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
-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
+Lemma. Let X_j = X_j^* lie in unital C*-algebras with states varphi_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.
+
+Proof. Let mu_j be the spectral law of X_j under varphi_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
+
+ int_{|x| > K} |x|^d, dmu_j(x) le C_r K^{d-2r} (K > 0).
+
+Take any weakly convergent subsequence with limit mu. Lower
+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
+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
-limit. This is the claimed convergence for $f$.
+limit. This is the claimed convergence for f.
-Proof of the proposition. Step 1, parameters. Fix $H, eps, m$. Choose
-$t > 0$ with $(1 + t^2)^{-1/4} < eps/4$, then choose $delta > 0$ and
-$0 < eta < 1/2$ with $tsqrt2(sqrtdelta + 2eta) < eps/4$. Amenability
-gives a nonempty finite $F subset G$ with
-$|hF ,△, F| < delta|F|$ for $h in H$. Set $n = |F|$ and
-$J = F cup ∪_{h in H} hF$. Finally choose a real polynomial
+Proof of the proposition. Step 1, parameters. Fix H, eps, m. Choose
+t > 0 with (1 + t^2)^{-1/4} < eps/4, then choose delta > 0 and
+0 < eta < 1/2 with tsqrt2(sqrtdelta + 2eta) < eps/4. Amenability
+gives a nonempty finite F subset G with
+|hF ,△, F| < delta|F| for h in H. Set n = |F| and
+J = F cup ∪_{h in H} hF. Finally choose a real polynomial
-$$ p(s) = sum_{r=1}^D c_r s^r, qquad sup_{0 le s le 1} |p(s) - √(s)| < eta/√(n). $$
+ p(s) = sum_{r=1}^D c_r s^r, sup_{0 le s le 1} |p(s) - √(s)| < eta/√(n).
-Such a polynomial exists: approximate $√(s)$ on $[0,1]$ by a real
-polynomial within $eta/(2sqrt n)$, then subtract its value at zero.
-Thus $t, delta, eta, F, n, J, p$ are fixed before $N to ∞$ and are
+Such a polynomial exists: approximate √(s) on [0,1] by a real
+polynomial within eta/(2sqrt n), then subtract its value at zero.
+Thus t, delta, eta, F, n, J, p are fixed before N to ∞ and are
not chosen from any trace or limiting covariance.
-Step 2, uniform Gaussian comparison. Let $R = (R_{g,g'})_{g,g' in J}$ be
+Step 2, uniform Gaussian comparison. Let R = (R_{g,g'})_{g,g' in J} be
any real positive semidefinite matrix with diagonal one. Choose unit
-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
+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
-$$ P_E = frac1n sum_{g in E} v_g v_g^{mathsf T}sfT, qquad B_E = p(P_E), qquad q_E = < W, B_E W > - Tr(B_E). $$
+ 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).
-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 $p(P_E) - P_E^{1/2}$
-vanishes on the kernel of $P_E$, and the scalar approximation gives
+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}
+vanishes on the kernel of P_E, and the scalar approximation gives
-$$ |B_E - P_E^{1/2}|_{HS}^2 le rank(P_E), eta^2/n le eta^2, qquad |P_E^{1/2}|_{HS}^2 = Tr(P_E) = 1, $$
+ |B_E - P_E^{1/2}|_{HS}^2 le rank(P_E), eta^2/n le eta^2, |P_E^{1/2}|_{HS}^2 = Tr(P_E) = 1,
-hence $1 - eta le |B_E|_{HS} le 1 + eta$.
+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
+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
lemma gives
-$$ |P_F^{1/2} - P_{hF}^{1/2}|_{HS}^2 le |P_F - P_{hF}|_1 le (|F ,△, hF|)/(n) < delta. $$
+ |P_F^{1/2} - P_{hF}^{1/2}|_{HS}^2 le |P_F - P_{hF}|_1 le (|F ,△, hF|)/(n) < delta.
With the rank control this yields
-$|B_F - B_{hF}|_{HS} le sqrtdelta + 2eta$. Both forms
-$q_F, q_{hF}$ are evaluated on the same $W$, so their difference is the
-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 $ell ge 1$,
+|B_F - B_{hF}|_{HS} le sqrtdelta + 2eta. Both forms
+q_F, q_{hF} are evaluated on the same W, so their difference is the
+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,
-$$ |mathbb EbbE exp(iell t q_F)| le (1 + 4ell^2 t^2 |B_F|_{HS}^2)^{-1/4} le (1 + t^2)^{-1/4}. $$
+ |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}.
-All these bounds hold for every such $R$, singular or not, and no
-translation invariance of $R$ has been used.
+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}^{mathsf T}sfT.
Taking quadratic forms and traces shows
-$$ q_E = sum_{r=1}^D (c_r)/(n^r) sum_{g_1, ..., g_r in E} R_{g_1,g_2} ... R_{g_{r-1},g_r} (xi_{g_1} xi_{g_r} - R_{g_r,g_1}), $$
+ 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}),
-with the empty product interpreted as $1$ when $r = 1$. Use the same
-prescription in $C$:
+with the empty product interpreted as 1 when r = 1. Use the same
+prescription in C:
-$$ widetilde Q_{N,E} = sum_{r=1}^D (c_r)/(n^r) sum_{g_1, ..., g_r in E} k_{N;g_1,g_2} ... k_{N;g_{r-1},g_r} (z_{N,g_1} z_{N,g_r} - k_{N;g_r,g_1}), $$
+ widetilde Q_{N,E} = sum_{r=1}^D (c_r)/(n^r) sum_{g_1, ..., g_r in E} k_{N;g_1,g_2} ... k_{N;g_{r-1},g_r} (z_{N,g_1} z_{N,g_r} - k_{N;g_r,g_1}),
-$$ Q_{N,E} = tfrac12(widetilde Q_{N,E} + widetilde Q_{N,E}^{,*}). $$
+ Q_{N,E} = tfrac12(widetilde Q_{N,E} + widetilde Q_{N,E}^{,*}).
Multiplication is in the displayed order. The empirical covariance array
need satisfy no positivity condition, because this is a finite polynomial in
-elements of $C$; square-root functional calculus was used only for $P_E$ in
+elements of C; square-root functional calculus was used only for P_E in
the Gaussian comparison. The relabeling
-$baralpha_l(z_{N,g}) = z_{N,lg}$, $baralpha_l(k_{N;g,h}) = k_{N;lg,lh}$
-carries each tuple in the sum for $F$ to its left translate in the sum for
-$hF$, preserving the order of all factors, and taking self-adjoint parts
-commutes with $baralpha_h$. Hence
+baralpha_l(z_{N,g}) = z_{N,lg}, baralpha_l(k_{N;g,h}) = k_{N;lg,lh}
+carries each tuple in the sum for F to its left translate in the sum for
+hF, preserving the order of all factors, and taking self-adjoint parts
+commutes with baralpha_h. Hence
-$$ Q_{N,hF} = baralpha_h(Q_{N,F}) qquad (h in H, N ge 1). $$
+ 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
+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
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
+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
-$$ |Q_{N_j,hF} - Q_{N_j,F}|_{2,varphi_j} → |q_{hF} - q_F|_{L^2}, $$
+ |Q_{N_j,hF} - Q_{N_j,F}|_{2,varphi_j} → |q_{hF} - q_F|_{L^2},
-$$ varphi_j(exp(i s Q_{N_j,F})) → mathbb EbbE exp(i s q_F) $$
+ varphi_j(exp(i s Q_{N_j,F})) → mathbb EbbE 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
-$|k_{N;g,g'}| le M^2$ and $|z_{N,g}| le sqrt N M$ give only
-$|Q_{N,E}| le (N+1) sum_r |c_r| M^{2r}$. Scalar moment convergence
+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
+|k_{N;g,g'}| le M^2 and |z_{N,g}| le sqrt N M give only
+|Q_{N,E}| le (N+1) sum_r |c_r| M^{2r}. Scalar moment convergence
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 $varphi$,
+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,
differentiation gives
-$$ (d)/(ds)(e^{i(1-s)tX} e^{istY}) = e^{i(1-s)tX}, it(Y - X), e^{istY}, $$
+ (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,varphi} le t|X - Y|_{2,varphi}.
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,varphi_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 ∞} |varphi_j(V_{N_j}^ell)| = |mathbb EbbE 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.
+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
-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
+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
+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
real positive semidefinite matrices with diagonal one, so a subsequence has
-a limiting covariance $R$, and the sequential estimates contradict the
+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 $varphi mapsto |varphi(V_N^ell)|$
+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)|
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
-both suprema in the proposition are at most $eps/2 < eps$, and
-$Q = Q_{N,F}$, $V = V_N$ complete the proof.
+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
+both suprema in the proposition are at most eps/2 < eps, and
+Q = Q_{N,F}, V = V_N complete the proof.
What remains is to pass from the approximately invariant unitaries of the
-proposition to exact relations in $A^omega$, and then to the projections
+proposition to exact relations in A^omega, and then to the projections
of equivariant uniform property Gamma. Both passages use one
coordinate-selection fact.
-Lemma. If $x = [(x_j)] in A^omega$ has a bounded representative, then
+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 mathcal T_calT_omega(A)} |x|_{2,tau} = lim_{j to omega} sup_{sigma in mathcal T(A)}calT(A) |x_j|_{2,sigma},
-$$ sup_{tau in mathcal T_calT_omega(A)} |tau(x)| = lim_{j to omega} sup_{sigma in mathcal T(A)}calT(A) |sigma(x_j)|. $$
+ sup_{tau in mathcal T_calT_omega(A)} |tau(x)| = lim_{j to omega} sup_{sigma in mathcal T(A)}calT(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 mathcal T(A)calT(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
satisfying each individual bound with positive slack form a set in
-$omega$; their intersection is again in $omega$, hence nonempty and
+omega; their intersection is again in omega, hence nonempty and
indeed contains arbitrarily large coordinates. The maximizing trace may
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 $C$
+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
-$$ tau_a(c) = tau(ac). $$
+ tau_a(c) = tau(ac).
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
-approximately invariant unitary $V = exp(i t Q)$ from the
+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
+approximately invariant unitary V = exp(i t Q) from the
preceding proposition gives
-$$ |tau(a V^ell)| < eps, tau(a) le eps qquad (0 le a le 1, tau in mathcal T_calT_omega(A), 1 le ell le m, tau(a) > 0), $$
+ |tau(a V^ell)| < eps, tau(a) le eps (0 le a le 1, tau in mathcal T_calT_omega(A), 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 $C$.
+with zero left side when tau(a) = 0. Approximate invariance passes to
+limit traces by restriction to C.
-Proposition. There is a unitary $w in (A^omega cap A')^{baralpha}$
+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_{mathbb T}bbT 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 mathcal T_calT_omega(A), and every
+continuous f, where mathbb TbbT = {z in mathbb CbbC : |z| = 1} and mu
is normalized Haar measure.
-Proof. Choose increasing finite sets $H_m subset G$ with union $G$ and a
-sequence $a_1, a_2, ...$ dense in the positive unit ball of $A$ with
-$a_1 = 1$. Apply the uniform-unitary proposition with
-$(H, eps, m) = (H_m, 1/m, m)$, and call the resulting exponential unitary
-$V^{(m)} in C$. It has a representative $(v^{(m)}_j)_j$ of unitaries in
-$A$: lift the self-adjoint exponent to a bounded self-adjoint sequence and
-exponentiate coordinatewise. The bounds for those lifts may depend on $m$,
+Proof. Choose increasing finite sets H_m subset G with union G and a
+sequence a_1, a_2, ... dense in the positive unit ball of A with
+a_1 = 1. Apply the uniform-unitary proposition with
+(H, eps, m) = (H_m, 1/m, m), and call the resulting exponential unitary
+V^{(m)} in C. It has a representative (v^{(m)}_j)_j of unitaries in
+A: lift the self-adjoint exponent to a bounded self-adjoint sequence and
+exponentiate coordinatewise. The bounds for those lifts may depend on m,
whereas the unitary representatives always have norm one.
-Since $V^{(m)}$ commutes with $A$, the coordinate-selection lemma, the
+Since V^{(m)} commutes with A, the coordinate-selection lemma, the
weighted small moments, and the invariance estimates allow a coordinate
-$j(m) ge m$ at which $w_m = v^{(m)}_{j(m)}$ satisfies
+j(m) ge m at which w_m = v^{(m)}_{j(m)} satisfies
-$$ |[w_m, a_r]|_{2,u} < 2/m quad (1 le r le m), $$
+ |[w_m, a_r]|_{2,u} < 2/m (1 le r le m),
-$$ |alpha_h(w_m) - w_m|_{2,u} < 2/m quad (h in H_m), $$
+ |alpha_h(w_m) - w_m|_{2,u} < 2/m (h in H_m),
-$$ sup_{sigma in mathcal T(A)}calT(A) |sigma(a_r w_m^ell)| < 2/m quad (1 le r, ell le m). $$
+ sup_{sigma in mathcal T(A)}calT(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
+a_r (V^{(m)})^ell. All coordinate tests are finite in number, and their
suprema already range over all traces.
-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 $a in A$, $tau in mathcal T_calT_omega(A)$,
-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 $mathbb T$bbT$ together with
-$|tau(a f(w))| le |a| |f|_∞$ proves it for every continuous $f$.
+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),
+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
+|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 $1$.
+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 $mathbb T$bbT$ 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
+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
+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
-$$ e_r := max_{1 le i le k} |f_i^{(r)} - chi_i|_{L^2(mu)} → 0, $$
+ e_r := max_{1 le i le k} |f_i^{(r)} - chi_i|_{L^2(mu)} → 0,
for instance by replacing each indicator near its two endpoints by linear
-ramps whose total lengths tend to zero. Fix $r$ and write $f_i = f_i^{(r)}$,
-$e = e_r$. Pointwise bounds and the triangle inequality give
+ramps whose total lengths tend to zero. Fix r and write f_i = f_i^{(r)},
+e = e_r. Pointwise bounds and the triangle inequality give
-$$ |f_i^2 - f_i|_{L^2(mu)} le e, qquad |f_i f_j|_{L^2(mu)} le 2e (i ne j), $$
+ |f_i^2 - f_i|_{L^2(mu)} le e, |f_i f_j|_{L^2(mu)} le 2e (i ne j),
-$$ Bigl|sum_i f_i - 1Bigr|_{L^2(mu)} le k e, qquad |int_{mathbb T}bbT f_i, dmu - frac1k| le e. $$
+ Bigl|sum_i f_i - 1Bigr|_{L^2(mu)} le k e, |int_{mathbb T}bbT f_i, dmu - frac1k| le e.
-The first uses $|f_i^2 - f_i| le |f_i - chi_i|$ almost everywhere; the
+The first uses |f_i^2 - f_i| le |f_i - chi_i| almost everywhere; the
second expands
-$f_i f_j - chi_i chi_j = (f_i - chi_i) f_j + chi_i (f_j - chi_j)$ and
-uses $chi_i chi_j = 0$ almost everywhere; the remaining ones follow from
-$sum_i chi_i = 1$ almost everywhere and the $L^1$-$L^2$ inequality for a
+f_i f_j - chi_i chi_j = (f_i - chi_i) f_j + chi_i (f_j - chi_j) and
+uses chi_i chi_j = 0 almost everywhere; the remaining ones follow from
+sum_i chi_i = 1 almost everywhere and the L^1-L^2 inequality for a
probability measure.
-Set $y_i^{(r)} = f_i^{(r)}(w)$. These are central, invariant positive
+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
+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
-every positive contraction $a in A$ it also gives
+the y_i^{(r)} uniformly over tau in mathcal T_calT_omega(A), 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_{mathbb T}bbT 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
-coordinate-selection lemma gives a coordinate $j(r) ge r$ such that the
-positive contractions $y_{i,r} = f_i^{(r)}(w_{j(r)})$ satisfy, for all
+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
+coordinate-selection lemma gives a coordinate j(r) ge r such that the
+positive contractions y_{i,r} = f_i^{(r)}(w_{j(r)}) satisfy, for all
indicated indices,
-$$ |y_{i,r}^2 - y_{i,r}|_{2,u} < e_r + 1/r, qquad |y_{i,r} y_{j,r}|_{2,u} < 2 e_r + 1/r (i ne j), $$
+ |y_{i,r}^2 - y_{i,r}|_{2,u} < e_r + 1/r, |y_{i,r} y_{j,r}|_{2,u} < 2 e_r + 1/r (i ne j),
-$$ Bigl|sum_i y_{i,r} - 1Bigr|_{2,u} < k e_r + 1/r, $$
+ Bigl|sum_i y_{i,r} - 1Bigr|_{2,u} < k e_r + 1/r,
-$$ |[y_{i,r}, a_s]|_{2,u} < 1/r quad (s le r), qquad |alpha_h(y_{i,r}) - y_{i,r}|_{2,u} < 1/r quad (h in H_r), $$
+ |[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),
-$$ sup_{sigma in mathcal T(A)}calT(A) |sigma(a_s y_{i,r}) - (sigma(a_s))/(k)| < e_r + 1/r quad (s le r). $$
+ 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).
-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
+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
approximations.
-Let $p_i = [(y_{i,r})_r] in A^omega$. The representatives are uniformly
+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
+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 mathcal T(A),calT(A),
-$$ lim_{r to omega} sigma_r(a_s y_{i,r}) = frac1k lim_{r to omega} sigma_r(a_s), $$
+ lim_{r to omega} sigma_r(a_s y_{i,r}) = frac1k lim_{r to omega} sigma_r(a_s),
-which is the defining trace relation for $a_s$ and every limit trace, and
-norm density with linearity extends it to all $a in A$. The projections
-therefore realize equivariant uniform property Gamma and sum to $1$.
+which is the defining trace relation for a_s and every limit trace, and
+norm density with linearity extends it to all a in A. The projections
+therefore realize equivariant uniform property Gamma and sum to 1.
Proof of the theorem. The trace hypotheses were checked earlier: exactness,
a tracial state, and faithfulness of every trace follow from nuclearity,
stable finiteness, simplicity, and the quasitrace theorem. The last
proposition supplies equivariant uniform property Gamma, so all hypotheses
-of the absorption criterion hold and $alpha$ and
-$beta = alpha ⊗ id_Z$ are cocycle conjugate. The exact
+of the absorption criterion hold and alpha and
+beta = alpha ⊗ id_Z are cocycle conjugate. The exact
orientation of the equations is fixed as follows.
-An isomorphism $Phi : A to B$ is automatically unital, being surjective
-between unital algebras. Put $gamma_g = Phi alpha_g Phi^{-1}$. If
+An isomorphism Phi : A to B is automatically unital, being surjective
+between unital algebras. Put gamma_g = Phi alpha_g Phi^{-1}. If
cocycle conjugacy is supplied in the form
-$gamma_g = Ad(u_g) beta_g$ for a $beta$-cocycle $u$, then the
+gamma_g = Ad(u_g) beta_g for a beta-cocycle u, then the
two displayed equations of the theorem follow directly. In the reverse
-orientation, suppose $beta_g = Ad(v_g) gamma_g$ with
-$v_{gh} = v_g gamma_g(v_h)$. The unitaries lie in $B$, whose multiplier
-algebra is $B$ itself. Set $u_g = v_g^*$. Then
+orientation, suppose beta_g = Ad(v_g) gamma_g with
+v_{gh} = v_g gamma_g(v_h). The unitaries lie in B, whose multiplier
+algebra is B itself. Set u_g = v_g^*. Then
-$$ u_g beta_g(u_h) = v_g^* v_g gamma_g(v_h^*) v_g^* = gamma_g(v_h^*) v_g^* = v_{gh}^* = u_{gh}, qquad Ad(u_g) beta_g = gamma_g. $$
+ u_g beta_g(u_h) = v_g^* v_g gamma_g(v_h^*) v_g^* = gamma_g(v_h^*) v_g^* = v_{gh}^* = u_{gh}, Ad(u_g) beta_g = gamma_g.
-Either orientation therefore gives the required $beta$-cocycle, and the
-identity relation $u_e = u_e^2$ together with unitarity forces $u_e = 1$.
+Either orientation therefore gives the required beta-cocycle, and the
+identity relation u_e = u_e^2 together with unitarity forces u_e = 1.
This yields both displayed equations and proves the theorem.