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
for every amenable action in the simple, nuclear, stably finite, unital
setting. Throughout, C*-algebras are complex, tensor products are
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 ~= 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 ~= 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,  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.

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 |-> 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
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
stable algebras are automatically equivariantly Jiang-Su stable; the
nonunital case lies outside the present argument. Earlier affirmative
cases cover finite algebras with finitely many rays of extremal traces and
very weak comparison.

pdf/szabo-1904.10897-equivariant-property-si.pdf

Earlier characterizations of Z-absorption invoke a finiteness restriction
on the trace simplex, which the present argument avoids: the Gaussian
splittings are uniform over the whole trace space rather than derived from
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
infinite-dimensional nuclear C*-algebra with finitely many extremal
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).

The decisive criterion reads:

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.

pdf/szabo-wouters-2301.12846-equivariant-property-gamma.pdf

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 T(B) write
|x|_{2,tau} = tau(x^*x)^{1/2} and
|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 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.

pdf/haagerup-1403.7653-quasitraces-are-traces.pdf

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
uniform tracial ultrapower

 A^omega = ell^∞(N, A) / I_omega,  I_omega = {(a_j) : lim_{j to omega} |a_j|_{2,u} = 0}. 

The ineualities |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 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 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'.

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 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
equivariant uniform property Gamma, is cocycle conjugate to its tensor
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
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.

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) 

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
T(C) != ∅. There are moreover self-adjoint elements
x_{i,g} in C_i and a constant M < ∞ with

 |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 ~= 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,

 |[theta_j(z), d_{s,j}]| le (2|z|)/(j). 

The coordinatewise homomorphisms therefore induce a unital *-homomorphism
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
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

 [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
separable, yielding the required algebras and commutation. A limit trace on
A^omega restricts to a tracial state on C, so
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 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.

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 phi is an extreme point of
T(B), then

 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_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 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 M by separate ultraweak
continuity, so M is a finite von Neumann algebra with faithful
normal trace tau.

Extremality forces M to be a factor. A nontrivial central
projection p in M would give the convex decomposition

 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 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 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 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 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 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 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
successively gives the product formula. This proves the lemma.

For N ge 1 and g, h in G set

 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

 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 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 phi_j to ((xi_g)_g, (R_{g,h})_{g,h}). Equivalently,
for every noncommutative *-polynomial P in these self-adjoint variables,

 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, phi_j) a subsequence satisfies the covariance
convergence.

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 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
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.

Consider a word of length d ge 1 in the fluctuation variables. At one
term of the sequence write N = N_j, phi = phi_j. Expanding the
sums gives

 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
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 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 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 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 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

 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} - phi_j(s_{i;g,h}) 1. 

The centering is row by row at the current trace, so
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} - 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 phi_j, and a singleton row again
vanishes by the independence lemma, so a surviving term uses at most
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

 |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 phi_j.

Finally replace each covariance factor in a general word by its scalar
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
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 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
|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

 sup_{phi in T(C)} |baralpha_h(V) - V|_{2,phi} < eps  (h in H), 

 sup_{phi in T(C)} |phi(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.

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). 

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
ineualities 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

 |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.

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
integral gives
|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.

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) 

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_{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
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.

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
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
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 phi_j,
and let q_B be a centered Gaussian quadratic. If
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 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

 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) = 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
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

 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
not chosen from any trace or limiting covariance.

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

 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}
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,  |P_E^{1/2}|_{HS}^2 = Tr(P_E) = 1, 

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^{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. 

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,

 |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}^{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}), 

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}), 

 Q_{N,E} = 1/2(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
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

 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 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 phi_j to the q_E. In particular

 |Q_{N_j,hF} - Q_{N_j,F}|_{2,phi_j} → |q_{hF} - q_F|_{L^2}, 

 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
|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 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 ineuality give the
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,phi_j} le tsqrt2(sqrtdelta + 2eta) < 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
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 (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 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 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.

What remains is to pass from the approximately invariant unitaries of the
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

 sup_{tau in T_} |x|_{2,tau} = lim_{j to omega} sup_{sigma in T(A)} |x_j|_{2,sigma}, 

 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 T(A) within 1/j of
the supremum for the test in question. Its limit trace realizes the reverse
ineuality, 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
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 T_, define on C

 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
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 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.

Proposition. There is a unitary w in (A^omega cap A')^{baralpha}
with

 tau(a f(w)) = tau(a) int_{T} f, dmu 

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
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
weighted small moments, and the invariance estimates allow a coordinate
j(m) ge m at which w_m = v^{(m)}_{j(m)} satisfies

 |[w_m, a_r]|_{2,u} < 2/m  (1 le r le m), 

 |alpha_h(w_m) - w_m|_{2,u} < 2/m  (h in H_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
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 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 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 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)} : T to [0,1] with

 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 ineuality 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_{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
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 ineuality for a
probability measure.

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 T_, and for
every positive contraction a in A it also gives

 |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
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,  |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, 

 |[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 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
approximations.

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 T(A),

 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.

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
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
cocycle conjugacy is supplied in the form
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

 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.
This yields both displayed equations and proves the theorem.