==== TABLE OF CONTENTS ====
Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//gaussian.tex
Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//introduction.tex
Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//paper.tex
Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//preliminaries.tex
Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//references.tex
Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//rows.tex
Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//splittings.tex
===========================

==== BEGIN Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//gaussian.tex ====
\section{Gaussian quadratics over F{\o}lner sets}
\label{sec:gaussian}

We retain the algebra \(C\), its invariant commuting orbit algebras
\(C_i\), and the variables \(x_{i,g}\), \(z_{N,g}\), and \(k_{N;g,h}\)
from Section~\ref{sec:rows}. In particular, \(\T(C)\) is nonempty,
the \(x_{i,g}\) are uniformly bounded by \(M\), and
Proposition~\ref{prop:joint-clt} applies to arbitrary sequences of
extreme traces. All traces of finite-dimensional matrices in this
section are unnormalized. For a matrix \(B\), we write
\(\|B\|_{\HS}=(\Tr(B^*B))^{1/2}\) and
\(\|B\|_1=\Tr|B|\).

\begin{proposition}[Uniform approximately invariant unitaries]\label{prop:uniform-unitaries}
For every finite set \(H\subset G\), every \(\eps>0\), and every
integer \(m\geq1\), there are \(t>0\) and a self-adjoint element
\(Q\in C\) such that \(V=\exp(\iu tQ)\) satisfies
\begin{align}
 \sup_{\varphi\in\T(C)}
 \|\bar\alpha_h(V)-V\|_{2,\varphi}
 &<\eps &&(h\in H), \label{eq:uniform-unitary-invariance}\\
 \sup_{\varphi\in\T(C)}|\varphi(V^\ell)|
 &<\eps &&(1\leq\ell\leq m).
 \label{eq:uniform-unitary-moments}
\end{align}
\end{proposition}

We first record the matrix and probability estimates used in the
proof.

\begin{lemma}[Square-root perturbation bound]\label{lem:square-root-trace}
If \(S,T\) are positive matrices of the same size, then
\[
 \|S^{1/2}-T^{1/2}\|_{\HS}^{\,2}\leq\|S-T\|_1.
\]
\end{lemma}

\begin{proof}
Set \(a=S^{1/2}\), \(b=T^{1/2}\), \(d=a-b\), and
\(u=\operatorname{sign}(d)\), with \(\operatorname{sign}(0)=0\).
Since \(a^2-b^2=ad+db\), cyclicity of the trace gives
\[
 \Tr\bigl(u(a^2-b^2)\bigr)
 =\Tr\bigl(|d|(a+b)\bigr).
\]
In a basis diagonalizing \(d\), positivity gives
\(a_{jj}+b_{jj}\geq|a_{jj}-b_{jj}|=|d_j|\). Thus
\[
 \Tr\bigl(|d|(a+b)\bigr)
 =\sum_j |d_j|(a_{jj}+b_{jj})
 \geq\sum_j d_j^2=\Tr(d^2).
\]
On the other hand, \(u\) is a contraction, so trace-norm duality
gives
\[
 \Tr\bigl(u(a^2-b^2)\bigr)\leq\Tr|a^2-b^2|.
\]
These two inequalities prove the assertion.
\end{proof}

\begin{lemma}[Gaussian quadratic estimates]\label{lem:gaussian-quadratic}
Let \(W\) be a standard Gaussian vector in a finite-dimensional
real Euclidean space, and let \(B\) be a real symmetric operator
on that space. Put
\[
 q_B=\langle W,BW\rangle-\Tr(B).
\]
Then
\begin{equation}\label{eq:quadratic-variance-characteristic}
 \|q_B\|_{L^2}=\sqrt{2}\,\|B\|_{\HS},
 \qquad
 \bigl|\mathbb E\exp(\iu s q_B)\bigr|
 \leq\bigl(1+4s^2\|B\|_{\HS}^2\bigr)^{-1/4}
 \quad(s\in\R).
\end{equation}
Its law has an exponential moment of its absolute value and is
determined by its moments.
\end{lemma}

\begin{proof}
Orthogonal diagonalization of \(B\) gives
\[
 q_B\ \overset{\mathrm{law}}{=}
 \sum_{j=1}^d\lambda_j(U_j^2-1),
\]
where \(\lambda_j\) are its eigenvalues and \(U_j\) are independent
standard real normal variables. The summands are centered and
have variances \(2\lambda_j^2\), proving the first formula.
The one-dimensional Gaussian integral gives
\[
 \bigl|\mathbb E\exp(\iu s q_B)\bigr|
 =\prod_{j=1}^d(1+4s^2\lambda_j^2)^{-1/4}.
\]
For nonnegative numbers \(a_j\), one has
\(\prod_j(1+a_j)\geq1+\sum_j a_j\). Applying this inequality with
\(a_j=4s^2\lambda_j^2\) proves the second formula in
Equation~\eqref{eq:quadratic-variance-characteristic}.

Choose \(c>0\) so that \(2c\max_j|\lambda_j|<1\); if \(B=0\),
any \(c>0\) will do. The pointwise estimate
\[
 \exp\!\left(c\left|\sum_j\lambda_j(U_j^2-1)\right|\right)
 \leq
 \exp\!\left(c\sum_j|\lambda_j|\right)
 \prod_j\exp(c|\lambda_j|U_j^2)
\]
and the Gaussian integral show that
\(\mathbb E\exp(c|q_B|)<\infty\).

For completeness, let \(\nu\) be another probability measure with
the same moments as \(q_B\). By the nonnegative power series for
\(\cosh\) and monotone convergence,
\[
 \int_{\R}\cosh(cx)\,d\nu(x)
 =\sum_{r=0}^{\infty}
   \frac{c^{2r}}{(2r)!}\mathbb E(q_B^{2r})
 =\mathbb E\cosh(cq_B)<\infty.
\]
Since \(e^{c|x|}\leq2\cosh(cx)\), the measure \(\nu\) also has an
exponential moment. The characteristic functions of \(\nu\) and
of \(q_B\) extend analytically to the strip
\(\{z\in\C:|\operatorname{Im}z|<c\}\). Their Taylor coefficients
at zero agree, so the Identity Theorem makes these analytic
functions equal. Uniqueness of characteristic functions proves
that \(\nu\) is the law of \(q_B\).
\end{proof}

The purpose of square-root normalization is already visible for a
positive real symmetric matrix \(P\) with \(\Tr(P)=1\).
Equation~\eqref{eq:quadratic-variance-characteristic} gives
\[
 \operatorname{Var}(q_{P^{1/2}})=2,
 \qquad
 \operatorname{Var}(q_P)=2\Tr(P^2).
\]
The latter variance is \(2\) when \(P\) is a rank-one projection,
but \(2/n\) when \(P=n^{-1}I_n\). The exact square root therefore
prevents the variance from vanishing as the rank grows.
Lemma~\ref{lem:square-root-trace} simultaneously controls its
change under perturbation. The proof below replaces this exact
square root by a polynomial with an explicitly controlled error.

\begin{lemma}[From moments to bounded continuous functions]\label{lem:quadratic-moment-transfer}
Let \(X_j=X_j^*\) belong to unital \(\mathrm{C}^*\)-algebras
equipped with states \(\varphi_j\). Let \(q_B\) be a centered
Gaussian quadratic as in Lemma~\ref{lem:gaussian-quadratic}.
Suppose that
\[
 \varphi_j(X_j^d)\longrightarrow\mathbb E(q_B^d)
 \qquad(d=0,1,2,\ldots).
\]
Then, for every bounded continuous function \(f:\R\to\C\),
\[
 \varphi_j(f(X_j))\longrightarrow\mathbb E f(q_B).
\]
No uniform bound on \(\|X_j\|\) is needed.
\end{lemma}

\begin{proof}
Let \(\mu_j\) be the spectral probability law of \(X_j\) under
\(\varphi_j\). The second moments of \(\mu_j\) are bounded, so
the sequence is tight. Fix an integer \(d\geq0\), and choose an
even integer \(2r>d\). Moment convergence supplies a constant
\(C_r<\infty\) such that
\(\int |x|^{2r}\,d\mu_j(x)\leq C_r\) for all \(j\). Consequently
\begin{equation}\label{eq:spectral-moment-tail}
 \int_{|x|>K}|x|^d\,d\mu_j(x)
 \leq C_r K^{d-2r}
 \qquad(K>0).
\end{equation}

Consider any weakly convergent subsequence, with limit \(\mu\).
Lower semicontinuity gives
\(\int |x|^{2r}\,d\mu(x)\leq C_r\), so the same tail bound holds
for \(\mu\). Multiply \(x^d\) by a continuous cutoff which equals
one on \([-K,K]\) and zero outside \([-2K,2K]\). Weak convergence
applies to this bounded continuous function. Letting \(K\) tend
to infinity and using Equation~\eqref{eq:spectral-moment-tail} shows that
\[
 \int x^d\,d\mu(x)=\mathbb E(q_B^d).
\]
This holds for every \(d\). Lemma~\ref{lem:gaussian-quadratic}
therefore identifies \(\mu\) with the law of \(q_B\).
Tightness now shows that the whole sequence \(\mu_j\) converges
weakly to that law: every subsequence has a weakly convergent
further subsequence, and all such limits have just been
identified. This is precisely the claimed convergence for \(f\).
\end{proof}

\begin{proof}[Proof of Proposition~\ref{prop:uniform-unitaries}]
\textbf{Step 1. Choice of parameters.}
Fix \(H,\eps,m\) as in the statement. Choose \(t>0\) so that
\begin{equation}\label{eq:gaussian-t-choice}
 (1+t^2)^{-1/4}<\eps/4.
\end{equation}
Next choose \(\delta>0\) and \(0<\eta<1/2\) so that
\begin{equation}\label{eq:gaussian-error-choice}
 t\sqrt{2}\,(\sqrt{\delta}+2\eta)<\eps/4.
\end{equation}
Amenability gives a nonempty finite \(F\subset G\) satisfying
\[
 |hF\mathbin{\triangle}F|<\delta|F|
 \qquad(h\in H).
\]
Set
\[
 n=|F|,
 \qquad J=F\cup\bigcup_{h\in H}hF.
\]
Finally, choose a real polynomial
\begin{equation}\label{eq:square-root-polynomial}
 p(s)=\sum_{r=1}^{D}c_rs^r,
 \qquad
 \sup_{0\leq s\leq1}|p(s)-\sqrt{s}|<\eta/\sqrt n.
\end{equation}
Such a polynomial exists: first approximate \(\sqrt{s}\) on
\([0,1]\) by a real polynomial to within
\(\eta/(2\sqrt n)\), then subtract its value at zero.
Thus \(t,\delta,\eta,F,n,J\), and \(p\), including its degree
and coefficients, are all fixed before \(N\) tends to infinity.
None is chosen using a trace or a limiting covariance.

\textbf{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 \(R\), spanning a finite-dimensional real Euclidean space
\(L\), and let \(W\) be standard Gaussian in \(L\). Write
\(\xi_g=\langle v_g,W\rangle\). For \(E=F\) or \(E=hF\), define
\[
 P_E=\frac1n\sum_{g\in E}v_gv_g^{\mathsf T},
 \qquad
 B_E=p(P_E),
 \qquad
 q_E=\langle W,B_EW\rangle-\Tr(B_E).
\]
The operator \(P_E\) is positive with trace one, so its spectrum
lies in \([0,1]\); its rank is at most \(n\).
Because \(p(0)=0\), the operator
\(p(P_E)-P_E^{1/2}\) vanishes on the kernel of \(P_E\).
The scalar approximation in Equation~\eqref{eq:square-root-polynomial}
therefore gives
\begin{equation}\label{eq:gaussian-rank-control}
 \|B_E-P_E^{1/2}\|_{\HS}^2
 \leq \operatorname{rank}(P_E)\,\eta^2/n
 \leq\eta^2,
 \qquad
 \|P_E^{1/2}\|_{\HS}^2=\Tr(P_E)=1.
\end{equation}
In particular,
\begin{equation}\label{eq:gaussian-nondegeneracy}
 1-\eta\leq\|B_E\|_{\HS}\leq1+\eta.
\end{equation}

The summands indexed by \(F\cap hF\) cancel when subtracting
\(P_F\) and \(P_{hF}\). Each remaining \(v_gv_g^{\mathsf T}\)
has trace norm one. Hence Lemma~\ref{lem:square-root-trace}
implies
\[
 \|P_F^{1/2}-P_{hF}^{1/2}\|_{\HS}^2
 \leq\|P_F-P_{hF}\|_1
 \leq\frac{|F\mathbin{\triangle}hF|}{n}<\delta.
\]
Combining this with Equation~\eqref{eq:gaussian-rank-control} yields
\[
 \|B_F-B_{hF}\|_{\HS}\leq\sqrt{\delta}+2\eta.
\]
Both forms \(q_F,q_{hF}\) are evaluated on the same \(W\), so
their difference is the centered quadratic associated with
\(B_{hF}-B_F\). Lemma~\ref{lem:gaussian-quadratic} gives
\begin{equation}\label{eq:gaussian-translation-bound}
 \|q_{hF}-q_F\|_{L^2}
 \leq\sqrt{2}\,(\sqrt{\delta}+2\eta).
\end{equation}
Also, Equation~\eqref{eq:gaussian-nondegeneracy} and \(\eta<1/2\)
give, for every integer \(\ell\geq1\),
\begin{equation}\label{eq:gaussian-fourier-bound}
 \bigl|\mathbb E\exp(\iu\ell t q_F)\bigr|
 \leq
 \bigl(1+4\ell^2t^2\|B_F\|_{\HS}^2\bigr)^{-1/4}
 \leq(1+t^2)^{-1/4}.
\end{equation}
All these bounds hold for every such \(R\), including singular
matrices. No invariance of \(R\) under translation of its indices
has been used.

\textbf{Step 3. Equivariant polynomial prescription.}
For \(r\geq1\), direct multiplication of the rank-one summands
gives
\[
 P_E^r
 =\frac1{n^r}
  \sum_{g_1,\ldots,g_r\in E}
  R_{g_1,g_2}\cdots R_{g_{r-1},g_r}
  v_{g_1}v_{g_r}^{\mathsf T}.
\]
Taking its quadratic form and its trace, respectively, shows that
\begin{equation}\label{eq:gaussian-covariance-polynomial}
 q_E=
 \sum_{r=1}^{D}\frac{c_r}{n^r}
 \sum_{g_1,\ldots,g_r\in E}
 R_{g_1,g_2}\cdots R_{g_{r-1},g_r}
 \bigl(\xi_{g_1}\xi_{g_r}-R_{g_r,g_1}\bigr).
\end{equation}
The product of covariance entries is interpreted as \(1\) when
\(r=1\).

Use the same prescription in \(C\) by setting
\begin{align}
 \widetilde Q_{N,E}
 &=
 \sum_{r=1}^{D}\frac{c_r}{n^r}
 \sum_{g_1,\ldots,g_r\in E}
 k_{N;g_1,g_2}\cdots k_{N;g_{r-1},g_r}
 \bigl(z_{N,g_1}z_{N,g_r}-k_{N;g_r,g_1}\bigr),
 \label{eq:empirical-quadratic-polynomial}\\
 Q_{N,E}
 &=\tfrac12\bigl(\widetilde Q_{N,E}
                 +\widetilde Q_{N,E}^{\,*}\bigr).
 \label{eq:empirical-quadratic-selfadjoint}
\end{align}
Multiplication in Equation~\eqref{eq:empirical-quadratic-polynomial} is
in the displayed order. The entries of the empirical covariance
array need satisfy no matrix positivity condition: this is a
finite polynomial in elements of \(C\), whereas square-root
functional calculus was applied only to \(P_E\) in the Gaussian
comparison.

The exact relabeling in Equation~\eqref{eq:empirical-equivariance} carries
each tuple in the sum for \(F\) to its left translate in the sum
for \(hF\). It preserves the order of
all factors, and taking self-adjoint parts commutes with
\(\bar\alpha_h\). Consequently
\begin{equation}\label{eq:quadratic-exact-equivariance}
 Q_{N,hF}=\bar\alpha_h(Q_{N,F})
 \qquad(h\in H,\ N\geq1).
\end{equation}

\textbf{Step 4. Transfer of moments and scalar laws.}
Consider now any sequence \(N_j\to\infty\) and extreme traces
\(\varphi_j\in\T(C)\) such that
\[
 \varphi_j(k_{N_j;g,g'})\longrightarrow R_{g,g'}
 \qquad(g,g'\in J).
\]
Proposition~\ref{prop:joint-clt} gives convergence of every
ordered mixed moment of the empirical variables. Any moment of
the polynomials in
Equations~\eqref{eq:empirical-quadratic-polynomial}--\eqref{eq:empirical-quadratic-selfadjoint}
is a finite linear combination of such moments. Since the
limiting variables commute and
Equation~\eqref{eq:gaussian-covariance-polynomial} is real-valued, it
follows that the \(Q_{N_j,E}\) converge jointly in moments under
\(\varphi_j\) to the \(q_E\). In particular,
\begin{equation}\label{eq:quadratic-difference-limit}
 \|Q_{N_j,hF}-Q_{N_j,F}\|_{2,\varphi_j}
 \longrightarrow \|q_{hF}-q_F\|_{L^2}.
\end{equation}
Lemma~\ref{lem:quadratic-moment-transfer} also gives
\begin{equation}\label{eq:quadratic-exponential-limit}
 \varphi_j\bigl(\exp(\iu s Q_{N_j,F})\bigr)
 \longrightarrow\mathbb E\exp(\iu s q_F)
 \qquad(s\in\R).
\end{equation}
This use of scalar spectral laws is pertinent because the
operator norms of the \(Q_{N,E}\) may grow with \(N\).
Indeed, the elementary estimates
\(\|k_{N;g,g'}\|\leq M^2\) and
\(\|z_{N,g}\|\leq\sqrt N\,M\) yield only
\[
 \|Q_{N,E}\|\leq(N+1)\sum_{r=1}^{D}|c_r|M^{2r}.
\]
Convergence of the scalar moments, together with the tail
estimate Equation~\eqref{eq:spectral-moment-tail}, is sufficient for
Equation~\eqref{eq:quadratic-exponential-limit}.

\textbf{Step 5. Uniformity over all traces.}
Set \(V_N=\exp(\iu tQ_{N,F})\). For any self-adjoint \(X,Y\)
in a unital \(\mathrm{C}^*\)-algebra and any tracial state
\(\varphi\), differentiation gives
\[
 \frac{d}{ds}
 \bigl(e^{\iu(1-s)tX}e^{\iu stY}\bigr)
 =e^{\iu(1-s)tX}\,\iu t(Y-X)\,e^{\iu stY}.
\]
The tracial \(2\)-seminorm is unchanged by left and right
unitary multiplication. Integration and the triangle inequality
therefore give
\begin{equation}\label{eq:tracial-duhamel}
 \|e^{\iu tX}-e^{\iu tY}\|_{2,\varphi}
 \leq t\|X-Y\|_{2,\varphi}.
\end{equation}
In view of Equations~\eqref{eq:quadratic-exact-equivariance},
\eqref{eq:quadratic-difference-limit}, and
\eqref{eq:gaussian-translation-bound}, every sequence of traces
and indices with convergent covariance as above satisfies
\begin{align}
 \limsup_{j\to\infty}
 \|\bar\alpha_h(V_{N_j})-V_{N_j}\|_{2,\varphi_j}
 &\leq t\sqrt{2}\,(\sqrt{\delta}+2\eta)<\eps/4,
 \label{eq:sequential-invariance-estimate}\\
 \lim_{j\to\infty}|\varphi_j(V_{N_j}^{\ell})|
 &=
 \bigl|\mathbb E\exp(\iu\ell t q_F)\bigr|
 \leq(1+t^2)^{-1/4}<\eps/4.
 \label{eq:sequential-fourier-estimate}
\end{align}
Here \(h\in H\) and \(1\leq\ell\leq m\), and the strict
inequalities are Equations~\eqref{eq:gaussian-error-choice} and
\eqref{eq:gaussian-t-choice}.

It follows that, for all sufficiently large \(N\), every extreme
trace \(\varphi\in\T(C)\) satisfies
\begin{equation}\label{eq:extreme-uniform-estimates}
 \|\bar\alpha_h(V_N)-V_N\|_{2,\varphi}<\eps/2
 \quad(h\in H),\qquad
 |\varphi(V_N^\ell)|<\eps/2
 \quad(1\leq\ell\leq m).
\end{equation}
Indeed, otherwise choose \(N_j\to\infty\) and an extreme trace
\(\varphi_j\) for which one of these finitely many tests fails.
Passing to a subsequence fixes the failed test. The matrices
\[
 \bigl(\varphi_j(k_{N_j;g,g'})\bigr)_{g,g'\in J}
\]
belong to the compact set of real positive semidefinite matrices
with diagonal one. A further subsequence has a limiting
covariance \(R\), and
Equation~\eqref{eq:sequential-invariance-estimate} or
Equation~\eqref{eq:sequential-fourier-estimate} contradicts the failure
at tolerance \(\eps/2\).
Only the finite covariance matrices were extracted; no
compactness of the extreme trace boundary is needed. Nor is a
convergence rate uniform in \(R\) required: each putative
sequence of failures supplies one limiting \(R\), to which the
preceding moment argument applies. Moment determinacy is used
after that \(R\) has been fixed, so no common exponential-moment
radius over all covariance matrices is required.

Fix an \(N\) for which Equation~\eqref{eq:extreme-uniform-estimates}
holds. For each \(h\in H\), the squared seminorm
\[
 \varphi\longmapsto
 \varphi\bigl((\bar\alpha_h(V_N)-V_N)^*
              (\bar\alpha_h(V_N)-V_N)\bigr)
\]
is weak-star continuous and affine on \(\T(C)\). For each
\(1\leq\ell\leq m\), the map
\(\varphi\mapsto|\varphi(V_N^\ell)|\) is weak-star continuous
and convex. The sublevel sets with bounds \(\eps^2/4\) and
\(\eps/2\), respectively, are closed and convex and contain
all extreme traces. By the Krein--Milman Theorem they contain
\(\T(C)\). Thus the suprema in
Equations~\eqref{eq:uniform-unitary-invariance} and
\eqref{eq:uniform-unitary-moments} are at most \(\eps/2\),
and in particular strictly less than \(\eps\).
Taking \(Q=Q_{N,F}\) and \(V=V_N\) completes the proof.
\end{proof}
==== END Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//gaussian.tex ====

==== BEGIN Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//introduction.tex ====
\section{Introduction}

Jiang--Su stability is a regularity property of a \(\mathrm C^*\)-algebra
that also has a dynamical counterpart: an action should absorb the
trivial action on the Jiang--Su algebra up to cocycle conjugacy.
The theorem below establishes this implication for all amenable actions
in the simple, nuclear, finite unital setting. Throughout, tensor products
are minimal, and \(\Zalg\) denotes the Jiang--Su algebra.

\begin{theorem}\label{thm:main}
Let \(A\) be a simple, separable, unital, infinite-dimensional, nuclear,
stably finite complex \(\mathrm C^*\)-algebra satisfying
\(A\cong A\otimes\Zalg\). Let \(G\) be a countable discrete amenable group
and let \(\alpha:G\to\Aut(A)\) be any action. Put
\[
 B=A\otimes\Zalg,\qquad
 \beta_g=\alpha_g\otimes\operatorname{id}_{\Zalg}.
\]
There are a unital \(*\)-isomorphism \(\Phi:A\to B\) and unitaries
\(u_g\in B\), \(g\in G\), such that
\begin{align}
 u_e&=1,& u_{gh}&=u_g\beta_g(u_h),\label{eq:cocycle}\\
 \Phi(\alpha_g(a))&=u_g\beta_g(\Phi(a))u_g^*
 &&(g\in G,\ a\in A).\label{eq:conjugacy}
\end{align}
\end{theorem}

Here stable finiteness means that every isometry in every matrix algebra
over \(A\) is unitary. Amenability is used in its F{\o}lner form: for
every finite \(H\subset G\) and \(\delta>0\), some nonempty finite
\(F\subset G\) satisfies
\(\lvert hF\mathbin{\triangle}F\rvert<\delta\lvert F\rvert\) for \(h\in H\).
Theorem~\ref{thm:main} imposes no outerness, Rokhlin, or UCT assumption,
and no condition on the extreme tracial boundary or on trace orbits.
In particular, it includes finite groups and trivial actions.

\subsection{Context and scope}

Szab\'o formulated automatic equivariant Jiang--Su stability for
amenable actions on separable simple nuclear Jiang--Su-stable algebras
as Conjecture~A of \cite{SzaboSI2021}.
Theorem~\ref{thm:main} resolves its unital, stably finite case.
The original formulation also includes nonunital algebras, which lie
beyond the scope of the present theorem. Earlier affirmative cases
include finite algebras with finitely many rays of extremal traces
and very weak comparison \cite[Theorem~C]{SzaboSI2021}.

Szab\'o and Wouters proved that equivariant uniform property Gamma
characterizes equivariant Jiang--Su stability in the setting used here
\cite[Corollary~C]{SzaboWouters2025}. Their theorem supplies the
absorption step of the proof. Our task is to construct the required
invariant central trace splittings for an arbitrary action.
These splittings are uniform over the entire trace space, even when
the action moves traces.

\subsection{Construction and proof structure}

The technical argument has three parts.
First, tensorial absorption supplies countably many commuting
\emph{orbit algebras} inside the uniform tracial central sequence algebra.
Under every extreme trace on the algebra they generate, distinct orbit
algebras are independent. This intrinsic independence is useful because
the traces under consideration need not extend to the ambient algebra.
We then prove a joint central limit statement for normalized sums and
their empirical covariances, allowing the extreme trace to vary with
the number of rows.

Second, the covariance is incorporated into a quadratic Gaussian
construction. For a F{\o}lner set \(F\), its Gram vectors define a
positive operator \(P_F\) of trace one. Taking a polynomial approximation
to \(P_F^{1/2}\) gives a quadratic form whose variance stays bounded
away from zero and whose change under translation is small.
This square-root normalization handles covariance matrices of arbitrary
rank, including highly correlated ones. Empirical covariances turn the
same formula into exactly equivariant noncommutative polynomials.
Compactness of the finite covariance space then promotes the scalar
moment limits to estimates uniform over every tracial state.

Finally, weighting traces by elements of \(A\) yields a central invariant
unitary with Haar distribution relative to \(A\). Two explicit
diagonal selections give this unitary and then equal trace projections.
The construction of the rows appears in Section~\ref{sec:rows}, the
uniform unitary estimates in Section~\ref{sec:gaussian}, and the
diagonal selections and absorption argument in Section~\ref{sec:splittings}.
All parts of this construction are proved below; the final absorption
theorem is stated separately in Theorem~\ref{thm:absorption}.

Several ingredients can be used separately under their stated hypotheses:
intrinsic independence for commuting rows
(Lemma~\ref{lem:factor-independence}), moment estimates under varying
tracial states (Proposition~\ref{prop:joint-clt}), and convergence of
scalar spectral laws against bounded continuous functions without
uniform operator-norm bounds (Lemma~\ref{lem:quadratic-moment-transfer}).
==== END Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//introduction.tex ====

==== BEGIN Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//paper.tex ====
\documentclass[11pt,a4paper]{article}
\usepackage[margin=28mm]{geometry}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{microtype}
\usepackage{xcolor}
\usepackage[colorlinks=true,linkcolor=blue!55!black,citecolor=blue!55!black,urlcolor=blue!55!black]{hyperref}
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Zalg}{\mathcal Z}
\newcommand{\N}{\mathbb N}
\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\T}{\operatorname{T}}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\HS}{\mathrm{HS}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Ad}{\operatorname{Ad}}
\newcommand{\U}{\mathcal U}
\newcommand{\eps}{\varepsilon}
\newcommand{\iu}{\mathrm{i}}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\dist}{dist}
\title{Equivariant Jiang--Su Stability\\for Amenable Actions\\in the Unital Stably Finite Case}
\author{OpenAI}
\date{October 5, 2026}
\hypersetup{pdfauthor={OpenAI},pdftitle={Equivariant Jiang-Su Stability for Amenable Actions in the Unital Stably Finite Case}}
\begin{document}
\maketitle
\begin{abstract}
Every action of a countable discrete amenable group on a simple,
separable, unital, infinite-dimensional, nuclear, stably finite complex
\(\mathrm C^*\)-algebra that is already Jiang--Su stable absorbs the
trivial action on the Jiang--Su algebra up to cocycle conjugacy.
No restriction is imposed on the action on the tracial-state simplex.
This proves the unital, stably finite case of Szab\'o's Conjecture~A
on automatic equivariant Jiang--Su stability.
\end{abstract}
\input{sections/introduction}
\input{sections/preliminaries}
\input{sections/rows}
\input{sections/gaussian}
\input{sections/splittings}
\input{references}
\end{document}
==== END Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//paper.tex ====

==== BEGIN Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//preliminaries.tex ====
\section{Traces and the absorption criterion}\label{sec:preliminaries}

For a unital \(\mathrm C^*\)-algebra \(D\), write \(\T(D)\) for its
tracial-state space. For \(\tau\in\T(D)\), set
\[
 \|x\|_{2,\tau}=\tau(x^*x)^{1/2}.
\]
Under the assumptions of Theorem~\ref{thm:main}, \(\T(A)\) is a
nonempty compact convex set. Indeed, nuclearity implies exactness,
and a unital stably finite exact algebra has a tracial state;
moreover its \(2\)-quasitraces are traces
\cite[Theorem~5.11 and Corollary~5.12]{Haagerup2014}.
Every tracial state on the simple algebra \(A\) is faithful:
its null space \(\{a:\tau(a^*a)=0\}\) is a closed two-sided ideal
that does not contain \(1\).

Fix a free ultrafilter \(\omega\) on \(\N=\{1,2,\ldots\}\). Define
\begin{equation}\label{eq:ultrapower}
 \|a\|_{2,u}=\sup_{\tau\in\T(A)}\|a\|_{2,\tau},\qquad
 A^\omega=\ell^\infty(\N,A)/I_\omega,\qquad
 I_\omega=\{(a_j):\lim_{j\to\omega}\|a_j\|_{2,u}=0\}.
\end{equation}
The inequalities
\(\|ab\|_{2,u}\leq\|a\|\|b\|_{2,u}\) and
\(\|ba\|_{2,u}\leq\|a\|\|b\|_{2,u}\) show that \(I_\omega\)
is a closed two-sided ideal. Faithfulness of traces embeds \(A\)
as the constant sequences in \(A^\omega\).
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 \(A^\omega\), since
\(\lvert\tau_j(a_j)\rvert\leq\|a_j\|_{2,\tau_j}\).
These are the \emph{limit traces}, denoted \(\T_\omega(A)\).
We do not identify them with all tracial states on \(A^\omega\).

Automorphisms permute \(\T(A)\) and therefore preserve
\(\|\cdot\|_{2,u}\). Applying \(\alpha_g\) to representative sequences
defines an action \(\bar\alpha:G\curvearrowright A^\omega\).
It preserves \(A^\omega\cap A'\). The quotient in
Equation~\eqref{eq:ultrapower} is also the quotient of the norm
ultrapower by its uniform trace-kernel ideal, since norm-null
sequences belong to \(I_\omega\).

\begin{definition}\label{def:gamma}
The action \(\alpha\) has \emph{equivariant uniform property Gamma}
if, for every integer \(k\geq2\), there are pairwise orthogonal
projections \(p_1,\ldots,p_k\in(A^\omega\cap A')^{\bar\alpha}\) with
\begin{equation}\label{eq:gamma}
 \tau(ap_i)=\frac{\tau(a)}{k}
 \qquad(a\in A,\ \tau\in\T_\omega(A),\ 1\leq i\leq k).
\end{equation}
\end{definition}

This is the unital simple formulation in
\cite[Definition~A]{SzaboWouters2025}.
Our construction also gives \(\sum_i p_i=1\).
We use the following established theorem.

\begin{theorem}[Szab\'o--Wouters]\label{thm:absorption}
Let a countable discrete amenable group act on a separable, unital,
simple, nuclear, \(\Zalg\)-stable \(\mathrm C^*\)-algebra with
nonempty tracial-state space. If the action has equivariant uniform
property Gamma, it is cocycle conjugate to its tensor product with
the trivial action on \(\Zalg\).
\end{theorem}

Theorem~\ref{thm:absorption} is the implication of
\cite[Corollary~C]{SzaboWouters2025} needed here.
Thus it remains to construct the projections required by Definition~\ref{def:gamma}.
The next two sections work inside a separable subalgebra \(C\) of
\(A^\omega\cap A'\). There we use \emph{all} traces in \(\T(C)\);
this will allow weighted restrictions of limit traces in
Section~\ref{sec:splittings}.
==== END Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//preliminaries.tex ====

==== BEGIN Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//references.tex ====
\begin{samepage}
\begin{thebibliography}{99}
\raggedright

\bibitem{Haagerup2014}
U.~Haagerup,
\emph{Quasitraces on exact \(C^*\)-algebras are traces},
C.~R. Math. Rep. Acad. Sci. Canada \textbf{36} (2014), no.~2--3,
67--92.
\href{https://arxiv.org/abs/1403.7653}{arXiv:1403.7653}.

\bibitem{SzaboSI2021}
G.~Szab\'o,
\emph{Equivariant property (SI) revisited},
Anal. PDE \textbf{14} (2021), no.~4, 1199--1232.
\href{https://doi.org/10.2140/apde.2021.14.1199}
{doi:10.2140/apde.2021.14.1199}.
\href{https://arxiv.org/abs/1904.10897v4}{arXiv:1904.10897v4}.

\bibitem{SzaboWouters2025}
G.~Szab\'o and L.~Wouters,
\emph{Equivariant property Gamma and the tracial local-to-global principle
for \(C^*\)-dynamics},
Anal. PDE \textbf{18} (2025), no.~6, 1385--1432.
\href{https://doi.org/10.2140/apde.2025.18.1385}
{doi:10.2140/apde.2025.18.1385}.
\href{https://arxiv.org/abs/2301.12846v3}{arXiv:2301.12846v3}.

\bibitem{TomsWinter2007}
A.~S. Toms and W.~Winter,
\emph{Strongly self-absorbing \(C^*\)-algebras},
Trans. Amer. Math. Soc. \textbf{359} (2007), no.~8, 3999--4029.
\href{https://arxiv.org/abs/math/0502211v2}{arXiv:math/0502211v2}.

\end{thebibliography}
\end{samepage}
==== END Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//references.tex ====

==== BEGIN Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//rows.tex ====
\section{Commuting orbit rows and joint moments}
\label{sec:rows}

Throughout this section, \(A\), \(G\), and \(\alpha\) satisfy the
hypotheses of Theorem~\ref{thm:main}. Only countability of \(G\) is
needed for the constructions in this section.

\begin{proposition}[Commuting orbit rows]
\label{prop:rows}
There are unital embeddings
\(\iota_i\colon\Zalg\to A^\omega\), \(i\geq 1\), for which the unital
orbit algebras
\[
 C_i=C^*\bigl(\bar\alpha_g(\iota_i(\Zalg)):g\in G\bigr)
\]
commute with \(A\) and commute pairwise. In particular,
\[
 C=C^*\Bigl(\bigcup_{i\geq 1}C_i\Bigr)
 \subseteq A^\omega\cap A'
\]
is unital, separable, and \(\bar\alpha\)-invariant, and
\(\T(C)\neq\varnothing\).

Moreover, there are self-adjoint elements \(x_{i,g}\in C_i\),
\(i\geq 1\), \(g\in G\), and a constant \(M<\infty\), such that
\begin{equation}
\label{eq:row-normalization}
 \|x_{i,g}\|\leq M,\qquad
 \varphi(x_{i,g})=0,\qquad
 \varphi(x_{i,g}^2)=1
 \quad\bigl(\varphi\in\T(C)\bigr),
\end{equation}
and
\begin{equation}
\label{eq:row-equivariance}
 \bar\alpha_h(x_{i,g})=x_{i,hg}
 \qquad(i\geq1,\ g,h\in G).
\end{equation}
\end{proposition}

\begin{proof}
The Jiang--Su algebra is simple, unital, uniquely tracial, and
isomorphic to its infinite tensor power; see
\cite[Example~1.14(iv) and Corollary~1.11]{TomsWinter2007}.
Write \(\rho\) for its unique tracial state. Since \(A\) is
\(\Zalg\)-stable, fix an identification
\[
 A\cong A\otimes\bigotimes_{l=1}^{\infty}\Zalg.
\]
We use the increasing finite tensor stages in this identification.

We first show that every norm-separable \(C^*\)-subalgebra
\(D\subseteq A^\omega\) admits a unital embedding
\begin{equation}
\label{eq:relative-tail-embedding}
 \Zalg\longrightarrow A^\omega\cap D'.
\end{equation}
Choose a norm-dense sequence \((d_s)_{s\geq1}\) in \(D\), and choose
bounded representative sequences
\(d_s=[(d_{s,j})_{j\geq1}]\).
For each \(j\), choose a finite tensor stage containing elements
within \(1/j\) in norm of
\(d_{1,j},\ldots,d_{j,j}\). Let
\(\theta_j\colon\Zalg\to A\) be the embedding into a tensor factor
strictly after this stage. For \(j\geq s\) and \(z\in\Zalg\),
\[
 \|[\theta_j(z),d_{s,j}]\|\leq \frac{2\|z\|}{j}.
\]
The coordinatewise homomorphisms therefore induce a unital
\(*\)-homomorphism \(z\mapsto[(\theta_j(z))_j]\) into
\(A^\omega\cap D'\). It is injective because \(\Zalg\) is simple
and the unit of \(A^\omega\) is nonzero. This proves
Equation~\eqref{eq:relative-tail-embedding}.

Set \(D_0=A\). Inductively, suppose that \(D_{i-1}\) is a separable,
unital, \(\bar\alpha\)-invariant subalgebra of \(A^\omega\) containing
\(A\) and the preceding orbit algebras. Apply
Equation~\eqref{eq:relative-tail-embedding} to obtain
\(\iota_i(\Zalg)\subseteq D_{i-1}'\), define \(C_i\) as in the
statement, and put
\[
 D_i=C^*(D_{i-1},C_i).
\]
For \(b\in\iota_i(\Zalg)\), \(d\in D_{i-1}\), and \(g\in G\),
\[
 [\bar\alpha_g(b),d]
 =\bar\alpha_g\bigl([b,\bar\alpha_{g^{-1}}(d)]\bigr)=0,
\]
since \(D_{i-1}\) is invariant. Thus \(C_i\) commutes with
\(D_{i-1}\). Countability of \(G\) preserves separability at each
stage. This gives the required algebras and their commutation
properties. A limit trace on \(A^\omega\) restricts to a tracial
state on \(C\), so \(\T(C)\) is nonempty.

The trace \(\rho\) is faithful by simplicity of \(\Zalg\).
Choose a non-scalar self-adjoint element \(c\in\Zalg\). Then
\(\rho((c-\rho(c)1)^2)>0\), so rescaling \(c-\rho(c)1\) gives
a self-adjoint \(b\in\Zalg\) with
\(\rho(b)=0\) and \(\rho(b^2)=1\). Set
\[
 x_{i,g}=\bar\alpha_g(\iota_i(b)),\qquad M=\|b\|.
\]
Every \(\varphi\in\T(C)\) restricts along each translated unital
embedding of \(\Zalg\) to \(\rho\). This proves
Equation~\eqref{eq:row-normalization}, and
Equation~\eqref{eq:row-equivariance} follows from the action law.
\end{proof}

The following independence statement is intrinsic to the algebra
generated by the rows. In particular, it applies to traces of \(C\)
that do not extend to the ambient ultrapower.

\begin{lemma}[Independence at extreme traces]
\label{lem:factor-independence}
Let \(B\) be a unital \(C^*\)-algebra generated by pairwise commuting
unital \(C^*\)-subalgebras \(B_i\), \(i\geq1\). If
\(\varphi\in\partial_{\mathrm e}\T(B)\), then
\[
 \varphi(b_1\cdots b_s)=\prod_{j=1}^s\varphi(b_j)
\]
whenever \(b_j\in B_{i_j}\) and the indices \(i_1,\ldots,i_s\)
are distinct.
\end{lemma}

\begin{proof}
Let \((H_\varphi,\pi,\Omega)\) be the tracial GNS representation
of \(\varphi\), and put
\[
 \mathcal M=\pi(B)'',\qquad
 \mathcal M_i=\pi(B_i)''.
\]
We first record the relevant properties of this representation,
allowing \(\varphi\) to have a nonzero kernel on \(B\).
On the dense subspace of vectors \([a]\), \(a\in B\), right
multiplication \(R_b[a]=[ab]\) is bounded with norm at most
\(\|b\|\), by traciality. It commutes with \(\pi(B)\), while
the vectors \(R_b\Omega=[b]\) are dense. If
\(T\in\mathcal M\) satisfies \(T\Omega=0\), then
\[
 TR_b\Omega=R_bT\Omega=0\qquad(b\in B),
\]
so \(T=0\). Thus \(\Omega\) is separating for \(\mathcal M\).
The normal vector state
\(\tau(T)=\langle T\Omega,\Omega\rangle\) is consequently
faithful: for \(T\geq0\), \(\tau(T)=0\) implies
\(T^{1/2}\Omega=0\). Its traciality extends from \(\pi(B)\)
to \(\mathcal M\) by separate ultraweak continuity. Hence
\(\mathcal M\) is a finite von Neumann algebra with faithful
normal tracial state \(\tau\).

Extremality of \(\varphi\) implies that \(\mathcal M\) is a factor.
Indeed, a nontrivial central projection \(p\in\mathcal M\) would
give the convex decomposition
\[
 \varphi(a)
 =\tau(p)\frac{\tau(p\pi(a))}{\tau(p)}
  +\tau(1-p)\frac{\tau((1-p)\pi(a))}{\tau(1-p)}.
\]
The two normalized functionals are tracial states on \(B\). They are distinct:
if, for instance, the first equalled \(\varphi\), equality of
the corresponding normal functionals would extend from
\(\pi(B)\) to \(\mathcal M\), and evaluation at \(p\) would
give \(1=\tau(p)\). This contradicts
\(0<\tau(p)<1\).

The algebras \(\mathcal M_i\) commute and generate \(\mathcal M\).
Every element of the center of \(\mathcal M_i\) therefore
commutes with all of \(\mathcal M\), so
\[
 Z(\mathcal M_i)\subseteq Z(\mathcal M)=\C1.
\]
The restriction of \(\tau\) makes each \(\mathcal M_i\) a finite
factor with a faithful normal tracial state.

Fix \(i\), and let \(y\geq0\) belong to the von Neumann algebra
generated by the other \(\mathcal M_j\)'s. It commutes with
\(\mathcal M_i\). The functional
\[
 \psi_y(x)=\tau(y^{1/2}xy^{1/2})=\tau(xy),
 \qquad x\in\mathcal M_i,
\]
is normal and positive: sandwiching by \(y^{1/2}\) preserves
suprema of bounded increasing nets of positive operators.
Commutation of \(y\) with \(\mathcal M_i\) makes it tracial,
and \(0\leq\psi_y\leq\|y\|\tau\).
The Radon--Nikodym Theorem for normal functionals on a finite
von Neumann algebra therefore gives a density
\(h\in\mathcal M_i\), \(0\leq h\leq\|y\|1\), with
\(\psi_y(x)=\tau(hx)\). Invariance of \(\psi_y\) under unitary
conjugation forces \(h\) to commute with every unitary in
\(\mathcal M_i\). Hence \(h\) is central and thus scalar.
Evaluating at the unit identifies that scalar as \(\tau(y)\).
It follows that
\[
 \tau(xy)=\tau(x)\tau(y).
\]
Linearity extends this identity to arbitrary \(y\) in the
algebra generated by the other rows. Applying it successively
proves the required product formula.
\end{proof}

We now fix the algebras and elements from
Proposition~\ref{prop:rows}. For \(N\geq1\) and \(g,h\in G\), set
\begin{equation}
\label{eq:empirical-variables}
 z_{N,g}=\frac1{\sqrt N}\sum_{i=1}^N x_{i,g},
 \qquad
 k_{N;g,h}=\frac1N\sum_{i=1}^N
 \frac{x_{i,g}x_{i,h}+x_{i,h}x_{i,g}}2.
\end{equation}
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
\begin{equation}
\label{eq:empirical-equivariance}
 \bar\alpha_l(z_{N,g})=z_{N,lg},
 \qquad
 \bar\alpha_l(k_{N;g,h})=k_{N;lg,lh}
 \quad(l,g,h\in G).
\end{equation}

\begin{proposition}[Joint ordered moment limit]
\label{prop:joint-clt}
Let \(J\subseteq G\) be nonempty and finite. Let \(N_j\to\infty\)
be a sequence of positive integers, and let
\(\varphi_j\in\partial_{\mathrm e}\T(C)\). Suppose that
\begin{equation}
\label{eq:covariance-convergence}
 \varphi_j(k_{N_j;g,h})\longrightarrow R_{g,h}
 \qquad(g,h\in J).
\end{equation}
Then \(R=(R_{g,h})_{g,h\in J}\) is a real positive semidefinite
matrix with diagonal entries one.
Choose unit vectors \(v_g\), \(g\in J\), spanning a
finite-dimensional real Euclidean space \(L\), with
\(\langle v_g,v_h\rangle=R_{g,h}\). If \(W\) is a standard
Gaussian vector in \(L\) and
\(\xi_g=\langle v_g,W\rangle\), then
\[
 \bigl((z_{N_j,g})_{g\in J},(k_{N_j;g,h})_{g,h\in J}\bigr)
 \longrightarrow
 \bigl((\xi_g)_{g\in J},(R_{g,h})_{g,h\in J}\bigr)
\]
in all ordered joint moments under \(\varphi_j\).
Equivalently, for every noncommutative \(*\)-polynomial \(P\)
in these self-adjoint variables,
\begin{equation}
\label{eq:joint-polynomial-limit}
 \varphi_j\!\left(
 P\bigl((z_{N_j,g})_g,(k_{N_j;g,h})_{g,h}\bigr)\right)
 \longrightarrow
 \mathbb E\,P\bigl((\xi_g)_g,(R_{g,h})_{g,h}\bigr).
\end{equation}
For every sequence \((N_j,\varphi_j)\) as above, a subsequence
satisfying Equation~\eqref{eq:covariance-convergence} exists.
\end{proposition}

\begin{proof}
For a tracial state \(\varphi\) and self-adjoint \(x,y\),
\[
 \varphi(xy)=\varphi(yx)
 =\overline{\varphi(xy)}.
\]
Thus all the second moments under consideration are real
and symmetric. For real scalars \(a_g\),
\[
 \sum_{g,h\in J}a_ga_h\varphi_j(k_{N_j;g,h})
 =
 \frac1{N_j}\sum_{i=1}^{N_j}
 \varphi_j\!\left(\left(\sum_{g\in J}a_gx_{i,g}\right)^2\right)
 \geq0.
\]
The diagonal entries are one by
Equation~\eqref{eq:row-normalization}. Cauchy--Schwarz bounds every
entry in absolute value by one. Consequently these covariance
matrices lie in a compact set of real symmetric matrices;
their limits have the asserted properties, and the final
subsequence assertion follows. A Gram representation of
the positive semidefinite matrix \(R\) supplies \(L\) and
the vectors \(v_g\).

The empty word is immediate. We first consider a word of
length \(d\geq1\) in the fluctuation
variables. At one term of the given sequence write
\(N=N_j\) and \(\varphi=\varphi_j\). Expanding the sums gives
\begin{equation}
\label{eq:fluctuation-expansion}
 \varphi(z_{N,g_1}\cdots z_{N,g_d})
 =N^{-d/2}
 \sum_{i_1,\ldots,i_d=1}^N
 \varphi(x_{i_1,g_1}\cdots x_{i_d,g_d}).
\end{equation}
Partition the positions \(1,\ldots,d\) according to equality
of their row labels \(i_a\). Since distinct rows commute,
we may group the factors belonging to each row, preserving
their order within that row. Lemma~\ref{lem:factor-independence}
then factors the expectation over the distinct rows.
A singleton block gives zero by
Equation~\eqref{eq:row-normalization}. Each remaining summand has
absolute value at most \(M^d\).

A partition with \(b\) blocks has at most \(N^b\) choices of
distinct row labels. Its contribution to
Equation~\eqref{eq:fluctuation-expansion} is therefore bounded by
\(M^dN^{b-d/2}\). Among partitions without singleton blocks,
\(b\leq\lfloor d/2\rfloor\). It follows that odd moments
tend to zero, and, when \(d=2q\), only partitions into \(q\)
pairs can have a nonzero limit.

Fix such a pairing, writing its pairs as
\((a_l,b_l)\), \(1\leq l\leq q\), with \(a_l<b_l\).
Its contribution is
\[
 N^{-q}
 \sum_{\substack{1\leq i_1,\ldots,i_q\leq N\\
                  i_1,\ldots,i_q\ \mathrm{distinct}}}
 \prod_{l=1}^q
 \varphi(x_{i_l,g_{a_l}}x_{i_l,g_{b_l}}).
\]
Dropping the distinctness condition changes this by
\(O_{d,M}(N^{-1})\): there are at most
\(\binom q2 N^{q-1}\) tuples with a collision, and each
product is bounded by \(M^{2q}\).
The unrestricted sum factors, and traciality identifies
its factors with the empirical covariances:
\[
 \prod_{l=1}^q
 \left(\frac1N\sum_{i=1}^N
 \varphi(x_{i,g_{a_l}}x_{i,g_{b_l}})\right)
 =
 \prod_{l=1}^q\varphi(k_{N;g_{a_l},g_{b_l}}).
\]
All error estimates depend only on \(d\) and \(M\), so they
hold uniformly for the moving traces \(\varphi_j\).
By Equation~\eqref{eq:covariance-convergence}, the limiting even
moment is the sum over pairings of the corresponding
products of entries of \(R\). This is the Gaussian moment:
the moment-generating function of \((\xi_g)_{g\in J}\) is
\[
 \mathbb E\exp\!\left(\sum_{g\in J}t_g\xi_g\right)
 =
 \exp\!\left(\frac12\sum_{g,h\in J}t_gt_hR_{g,h}\right),
\]
whose Taylor expansion gives precisely this pairing formula.

It remains to include the covariance variables in arbitrary
positions of a word. For \(g,h\in J\), define
\[
 s_{i;g,h}=\frac{x_{i,g}x_{i,h}+x_{i,h}x_{i,g}}2,
 \qquad
 y_{j,i;g,h}=s_{i;g,h}-\varphi_j(s_{i;g,h})1.
\]
This is centering separately in each row, at the current
trace. In particular,
\[
 \varphi_j(y_{j,i;g,h})=0,\qquad
 \|y_{j,i;g,h}\|\leq2M^2,
\]
and there is the exact identity
\begin{equation}
\label{eq:rowwise-covariance-centering}
 k_{N_j;g,h}-\varphi_j(k_{N_j;g,h})1
 =\frac1{N_j}\sum_{i=1}^{N_j}y_{j,i;g,h}.
\end{equation}

Let \(U_j\) be an ordered word having \(d\) fluctuation factors
and \(r\geq1\) factors of the form on the left of
Equation~\eqref{eq:rowwise-covariance-centering}. Expand all its sums.
Each of its \(d+r\) row factors is centered under
\(\varphi_j\), and a singleton row again gives zero by
Lemma~\ref{lem:factor-independence}. A surviving term therefore
uses at most \(\lfloor(d+r)/2\rfloor\) distinct row labels.
The products within any one row have absolute expectation
bounded by the product of the operator norms of their
factors. There are only finitely many partitions of the
\(d+r\) positions. Together with the normalization
\(N_j^{-d/2-r}\), these observations give
\begin{equation}
\label{eq:centered-mixed-word-bound}
 \left|\varphi_j(U_j)\right|
 \leq C_{d,r,M}\,
 N_j^{\lfloor(d+r)/2\rfloor-d/2-r}
 \leq C_{d,r,M}\,N_j^{-r/2}
 \longrightarrow0,
\end{equation}
where the constant is 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
in Equation~\eqref{eq:rowwise-covariance-centering}. Every term
containing a centered covariance factor tends to zero by
Equation~\eqref{eq:centered-mixed-word-bound}. The remaining terms
converge by the fluctuation calculation and
Equation~\eqref{eq:covariance-convergence}. This proves convergence
of every ordered word. Since all generators are
self-adjoint, linearity also gives the
\(*\)-polynomial formulation in
Equation~\eqref{eq:joint-polynomial-limit}.
\end{proof}
==== END Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//rows.tex ====

==== BEGIN Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//splittings.tex ====
\section{Invariant trace splittings and absorption}\label{sec:splittings}

We pass from Proposition~\ref{prop:uniform-unitaries} to exact
relations in \(A^\omega\). We first record the coordinate-selection
fact used in both passages.

\begin{lemma}[Coordinate selection]\label{lem:coordinate}
If \(x=[(x_j)]\in A^\omega\) has a bounded representative, then
\begin{align}
 \sup_{\tau\in\T_\omega(A)}\|x\|_{2,\tau}
   &=\lim_{j\to\omega}\sup_{\sigma\in\T(A)}\|x_j\|_{2,\sigma},
       \label{eq:coordinate-two}\\
 \sup_{\tau\in\T_\omega(A)}|\tau(x)|
   &=\lim_{j\to\omega}\sup_{\sigma\in\T(A)}|\sigma(x_j)|.
       \label{eq:coordinate-trace}
\end{align}
Consequently, finitely many such tests can be realized simultaneously
at a single coordinate with any prescribed positive slack above their
ultralimit bounds.
\end{lemma}

\begin{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 inequality; 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 be different for every test.
\end{proof}

\subsection{A weighted Haar unitary}

Keep the invariant separable algebra \(C\subset A^\omega\cap A'\)
from Proposition~\ref{prop:rows}. For \(a\in A_+\) and
\(\tau\in\T_\omega(A)\), define on \(C\)
\begin{equation}\label{eq:weighted-trace}
 \tau_a(c)=\tau(ac).
\end{equation}
This is a positive tracial functional. Positivity follows from
\(ac=a^{1/2}ca^{1/2}\) when \(c\geq0\), since \(a\) commutes with \(C\).
For \(c,d\in C\), traciality and the same commutation give
\(\tau(acd)=\tau(dac)=\tau(adc)\). Its norm is \(\tau(a)\).
If \(\tau(a)>0\), its normalization is therefore an element of
\(\T(C)\); if \(\tau(a)=0\), the functional is zero.
Proposition~\ref{prop:uniform-unitaries} thus gives, for its unitary \(V\),
\begin{equation}\label{eq:weighted-small-moments}
 |\tau(aV^\ell)|<\eps\,\tau(a)\leq\eps
 \quad
 \bigl(0\leq a\leq1,\ \tau\in\T_\omega(A),\
       1\leq\ell\leq m,\ \tau(a)>0\bigr),
\end{equation}
with zero left side when \(\tau(a)=0\).
The approximate invariance estimates apply to limit traces by
restriction to \(C\).

\begin{proposition}[A weighted Haar unitary]\label{prop:haar}
There is a unitary \(w\in(A^\omega\cap A')^{\bar\alpha}\) such that,
for every \(a\in A\), \(\tau\in\T_\omega(A)\), and
\(f\in C(\mathbb T)\),
\begin{equation}\label{eq:haar}
 \tau(af(w))=\tau(a)\int_{\mathbb T}f\,d\mu,
\end{equation}
where \(\mathbb T=\{z\in\C:|z|=1\}\) and \(\mu\) is normalized
Haar measure.
\end{proposition}

\begin{proof}
Choose increasing finite sets \(H_m\subset G\) whose union is \(G\),
and a sequence \(a_1,a_2,\ldots\) dense in the positive unit ball of
\(A\), with \(a_1=1\). Apply
Proposition~\ref{prop:uniform-unitaries} with
\((H,\eps,m)=(H_m,1/m,m)\), and denote its exponential unitary by
\(V^{(m)}\in C\).
It has a representative \((v^{(m)}_j)_j\) consisting of unitaries
in \(A\): lift its self-adjoint exponent to a bounded self-adjoint
sequence and exponentiate coordinatewise. The bounds for these
self-adjoint lifts may depend on \(m\), whereas the unitary
representatives always have norm one.

Since \(V^{(m)}\) commutes with \(A\), Lemma~\ref{lem:coordinate},
Equation~\eqref{eq:weighted-small-moments}, and the invariance
estimates allow a coordinate \(j(m)\geq m\) at which
\(w_m=v^{(m)}_{j(m)}\) satisfies
\begin{align}
 \|[w_m,a_r]\|_{2,u}&<2/m &&(1\leq r\leq m),\label{eq:first-diagonal}\\
 \|\alpha_h(w_m)-w_m\|_{2,u}&<2/m &&(h\in H_m),\notag\\
 \sup_{\sigma\in\T(A)}|\sigma(a_r w_m^\ell)|
   &<2/m &&(1\leq r,\ell\leq m).\notag
\end{align}
For the last line, apply Equation~\eqref{eq:coordinate-trace} 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 lines of
Equation~\eqref{eq:first-diagonal}, density, and
\(\|[w_m,a-a_r]\|_{2,u}\leq2\|a-a_r\|\) show that it is central
relative to \(A\) and fixed by \(\bar\alpha\).
The last line, first for every \(a_r\), then by norm density for
positive contractions, and then by linearity for \(a\in A\), gives
\[
 \tau(aw^\ell)=0
 \qquad(a\in A,\ \tau\in\T_\omega(A),\ \ell\geq1).
\]
The same holds for negative powers, because
\(\tau(aw^{-\ell})=\overline{\tau(a^*w^\ell)}\).
The zeroth power has trace \(\tau(a)\).
Thus Equation~\eqref{eq:haar} holds for trigonometric polynomials.
Uniform approximation on \(\mathbb T\) and
\(|\tau(a f(w))|\leq\|a\|\|f\|_\infty\) prove it for every continuous
\(f\).
\end{proof}

\subsection{Equal trace projections}

\begin{proposition}[Equal trace projections]\label{prop:gamma}
The action \(\alpha\) has equivariant uniform property Gamma.
The projections in Definition~\ref{def:gamma} can be chosen to sum to \(1\).
\end{proposition}

\begin{proof}
Fix \(k\geq2\) and let \(w=[(w_j)]\) be the unitary of
Proposition~\ref{prop:haar}, represented by unitaries \(w_j\in A\).
Partition \(\mathbb T\) into \(k\) half-open arcs \(I_1,\ldots,I_k\)
of Haar measure \(1/k\), and write \(\chi_i=1_{I_i}\).
Choose continuous functions \(f_i^{(r)}:\mathbb T\to[0,1]\) satisfying
\[
 e_r:=\max_{1\leq i\leq k}\|f_i^{(r)}-\chi_i\|_{L^2(\mu)}
   \longrightarrow0.
\]
For example, replace each indicator near its two endpoints by linear
ramps, with the total lengths of the transition arcs tending to zero.
For this proof fix \(r\), and write \(f_i=f_i^{(r)}\) and \(e=e_r\).
Pointwise bounds and the triangle inequality imply
\begin{align}
 \|f_i^2-f_i\|_{L^2(\mu)}&\leq e,\label{eq:arc-bounds}\\
 \|f_i f_j\|_{L^2(\mu)}&\leq2e &&(i\ne j),\notag\\
 \Big\|\sum_i f_i-1\Big\|_{L^2(\mu)}&\leq ke,\notag\\
 \Big|\int_{\mathbb T} f_i\,d\mu-\frac1k\Big|&\leq e.\notag
\end{align}
For the first inequality, use
\(|f_i^2-f_i|\leq|f_i-\chi_i|\) almost everywhere.
For the second, expand
\(f_i f_j-\chi_i\chi_j=(f_i-\chi_i)f_j+\chi_i(f_j-\chi_j)\)
and use \(\chi_i\chi_j=0\) almost everywhere.
The remaining inequalities 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 contractions, since continuous
functional calculus respects commutation and automorphisms.
Equation~\eqref{eq:haar} with \(a=1\), applied to the squared modulus
of each continuous function in Equation~\eqref{eq:arc-bounds},
gives the corresponding \(2\)-norm bounds for \(y_i^{(r)}\),
uniformly over \(\tau\in\T_\omega(A)\).
For every positive contraction \(a\in A\), it also gives
\begin{equation}\label{eq:arc-weighted}
 \left|\tau(a y_i^{(r)})-\frac{\tau(a)}k\right|
   =\tau(a)\left|\int_{\mathbb T}f_i^{(r)}\,d\mu-\frac1k\right|
   \leq e_r .
\end{equation}

Represent \(y_i^{(r)}\) by \(f_i^{(r)}(w_j)\).
Use \(H_r\) and \(a_1,\ldots,a_r\) from the proof of
Proposition~\ref{prop:haar}.
For this fixed \(r\), Lemma~\ref{lem:coordinate} gives a coordinate
\(j(r)\geq r\) such that the positive contractions
\(y_{i,r}=f_i^{(r)}(w_{j(r)})\) satisfy, simultaneously for all
indicated indices,
\begin{align}
 \|y_{i,r}^2-y_{i,r}\|_{2,u}&<e_r+1/r,\label{eq:second-diagonal}\\
 \|y_{i,r}y_{j,r}\|_{2,u}&<2e_r+1/r &&(i\ne j),\notag\\
 \Big\|\sum_i y_{i,r}-1\Big\|_{2,u}&<ke_r+1/r,\notag\\
 \|[y_{i,r},a_s]\|_{2,u}&<1/r &&(s\leq r),\notag\\
 \|\alpha_h(y_{i,r})-y_{i,r}\|_{2,u}&<1/r &&(h\in H_r),\notag\\
 \sup_{\sigma\in\T(A)}
 \left|\sigma(a_s y_{i,r})-\frac{\sigma(a_s)}k\right|
   &<e_r+1/r &&(s\leq r).\notag
\end{align}
This is a coordinate choice after \(f_i^{(r)}\) has been fixed.
It therefore needs no uniform continuity modulus, as \(r\) varies,
for these increasingly sharp arc approximations.

Let \(p_i=[(y_{i,r})_r]\in A^\omega\).
Their representatives are uniformly bounded positive contractions.
The first three lines of Equation~\eqref{eq:second-diagonal} imply
that \(p_i\) are orthogonal projections summing to \(1\); the next
two lines imply centrality and invariance by density.
For every sequence \((\sigma_r)\) in \(\T(A)\), the last line gives
\[
 \lim_{r\to\omega}\sigma_r(a_s y_{i,r})
   =\frac1k\lim_{r\to\omega}\sigma_r(a_s).
\]
This is Equation~\eqref{eq:gamma} for \(a_s\), for every limit trace.
Norm density and linearity extend it to all \(a\in A\).
Thus the projections satisfy Definition~\ref{def:gamma}.
\end{proof}

\subsection{The cocycle equations}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
The trace hypotheses were checked in Section~\ref{sec:preliminaries},
and Proposition~\ref{prop:gamma} supplies equivariant uniform property
Gamma. All assumptions of Theorem~\ref{thm:absorption} hold.
It follows that \(\alpha\) and
\(\beta=\alpha\otimes\operatorname{id}_{\Zalg}\) are cocycle conjugate.
We spell out the orientation to obtain the exact equations of the
theorem.

An isomorphism \(\Phi:A\to B\) is automatically unital, since it is
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
Equations~\eqref{eq:cocycle} and~\eqref{eq:conjugacy} follow directly.
In the reverse orientation, suppose
\[
 \beta_g=\Ad(v_g)\gamma_g,\qquad
 v_{gh}=v_g\gamma_g(v_h).
\]
Here the unitaries lie in \(B\), whose multiplier algebra is \(B\)
itself. Set \(u_g=v_g^*\). Then
\[
 u_g\beta_g(u_h)
   =v_g^*v_g\gamma_g(v_h^*)v_g^*
   =\gamma_g(v_h^*)v_g^*
   =v_{gh}^*
   =u_{gh},
 \qquad
 \Ad(u_g)\beta_g=\gamma_g.
\]
Thus either orientation gives the required \(\beta\)-cocycle.
Its identity relation \(u_e=u_e^2\), together with unitarity,
forces \(u_e=1\). This proves both displayed equations and hence
the theorem.
\end{proof}
==== END Equivariant-Jiang-Su-Stability-for-Amenable-Actions-in-the-Unital-Stably-Finite-Case//splittings.tex ====