Equivariant Jiang–Su Stability for Amenable Actions in the Unital Stably Finite Case

Proof. The Jiang–Su algebra is simple, unital, uniquely tracial, and isomorphic to its infinite tensor power; see [4, Example 1.14(iv) and Corollary 1.11]. Write \rho for its unique tracial state. Since A is \Zalg-stable, fix an identification

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

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,

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 (3.3).

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 (3.3) to obtain \iota_i(\Zalg)\subseteq D_{i-1}', define C_i as in the statement, and put

For b\in\iota_i(\Zalg), d\in D_{i-1}, and g\in G,

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

Every \varphi\in\T(C) restricts along each translated unital embedding of \Zalg to \rho. This proves Equation (3.1), and Equation (3.2) follows from the action law.□

Proof. Let (H_\varphi,\pi,\Omega) be the tracial GNS representation of \varphi, and put

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

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

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

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

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

Linearity extends this identity to arbitrary y in the algebra generated by the other rows. Applying it successively proves the required product formula.□

Proof. For a tracial state \varphi and self-adjoint x,y,

Thus all the second moments under consideration are real and symmetric. For real scalars a_g,

The diagonal entries are one by Equation (3.1). 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

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 3.2 then factors the expectation over the distinct rows. A singleton block gives zero by Equation (3.1). 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 (3.8) 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

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:

All error estimates depend only on d and M, so they hold uniformly for the moving traces \varphi_j. By Equation (3.6), 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

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

This is centering separately in each row, at the current trace. In particular,

and there is the exact identity

Let U_j be an ordered word having d fluctuation factors and r\geq1 factors of the form on the left of Equation (3.9). 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 3.2. 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

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 (3.9). Every term containing a centered covariance factor tends to zero by Equation (3.10). The remaining terms converge by the fluctuation calculation and Equation (3.6). This proves convergence of every ordered word. Since all generators are self-adjoint, linearity also gives the *-polynomial formulation in Equation (3.7).□

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

In a basis diagonalizing d, positivity gives a_{jj}+b_{jj}\geq|a_{jj}-b_{jj}|=|d_j|. Thus

On the other hand, u is a contraction, so trace-norm duality gives

These two inequalities prove the assertion.□

Proof. Orthogonal diagonalization of B gives

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

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 (4.3).

Choose c>0 so that 2c\max_j|\lambda_j|<1; if B=0, any c>0 will do. The pointwise estimate

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,

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.□

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

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 (4.4) shows that

This holds for every d. Lemma 4.3 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.□

Proof of Proposition 4.1. Step 1. Choice of parameters. Fix H,\eps,m as in the statement. Choose t>0 so that

Next choose \delta>0 and 0<\eta<1/2 so that

Amenability gives a nonempty finite F\subset G satisfying

Set

Finally, choose a real polynomial

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.

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

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 (4.7) therefore gives

In particular,

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 4.2 implies

Combining this with Equation (4.8) yields

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 4.3 gives

Also, Equation (4.9) and \eta<1/2 give, for every integer \ell\geq1,

All these bounds hold for every such R, including singular matrices. No invariance of R under translation of its indices has been used.

Step 3. Equivariant polynomial prescription. For r\geq1, direct multiplication of the rank-one summands gives

Taking its quadratic form and its trace, respectively, shows that

The product of covariance entries is interpreted as 1 when r=1.

Use the same prescription in C by setting

Multiplication in Equation (4.13) 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 (3.5) 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

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

Proposition 3.3 gives convergence of every ordered mixed moment of the empirical variables. Any moment of the polynomials in Equations (4.13)–(4.14) is a finite linear combination of such moments. Since the limiting variables commute and Equation (4.12) is real-valued, it follows that the Q_{N_j,E} converge jointly in moments under \varphi_j to the q_E. In particular,

Lemma 4.4 also gives

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

Convergence of the scalar moments, together with the tail estimate Equation (4.4), is sufficient for Equation (4.17).

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

The tracial 2-seminorm is unchanged by left and right unitary multiplication. Integration and the triangle inequality therefore give

In view of Equations (4.15), (4.16), and (4.10), every sequence of traces and indices with convergent covariance as above satisfies

Here h\in H and 1\leq\ell\leq m, and the strict inequalities are Equations (4.6) and (4.5).

It follows that, for all sufficiently large N, every extreme trace \varphi\in\T(C) satisfies

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

belong to the compact set of real positive semidefinite matrices with diagonal one. A further subsequence has a limiting covariance R, and Equation (4.19) or Equation (4.20) 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 (4.21) holds. For each h\in H, the squared seminorm

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 (4.1) and (4.2) are at most \eps/2, and in particular strictly less than \eps. Taking Q=Q_{N,F} and V=V_N completes the proof.□

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.□

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 4.1 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 5.1, Equation (5.4), and the invariance estimates allow a coordinate j(m)\geq m at which w_m=v^{(m)}_{j(m)} satisfies

For the last line, apply Equation (5.2) 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 (5.6), 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

The same holds for negative powers, because \tau(aw^{-\ell})=\overline{\tau(a^*w^\ell)}. The zeroth power has trace \tau(a). Thus Equation (5.5) holds for trigonometric polynomials. Uniform approximation on \mathbb T and |\tau(a f(w))|\leq\|a\|\|f\|_\infty prove it for every continuous f.□

Proof. Fix k\geq2 and let w=[(w_j)] be the unitary of Proposition 5.2, 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

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

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 (5.5) with a=1, applied to the squared modulus of each continuous function in Equation (5.7), 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

Represent y_i^{(r)} by f_i^{(r)}(w_j). Use H_r and a_1,\ldots,a_r from the proof of Proposition 5.2. For this fixed r, Lemma 5.1 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,

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 (5.9) 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

This is Equation (2.2) for a_s, for every limit trace. Norm density and linearity extend it to all a\in A. Thus the projections satisfy Definition 2.1.□

Proof of Theorem 1.1. The trace hypotheses were checked in Section 2, and Proposition 5.3 supplies equivariant uniform property Gamma. All assumptions of Theorem 2.2 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 (1.1) and (1.2) follow directly. In the reverse orientation, suppose

Here the unitaries lie in B, whose multiplier algebra is B itself. Set u_g=v_g^*. Then

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.□