Proof. The Jiang–Su algebra is simple, unital, uniquely tracial, and isomorphic to its infinite tensor power; see [4]. Write \(\rho\) for its unique tracial state. Since \(A\) is \(\mathcal Z\)-stable, fix an identification \[A\cong A\otimes\bigotimes_{l=1}^{\infty}\mathcal Z.\] 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 \[ \mathcal Z\longrightarrow A^\omega\cap D'. \tag{7}\] 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 \(\theta_j\colon\mathcal Z\to A\) be the embedding into a tensor factor strictly after this stage. For \(j\geq s\) and \(z\in\mathcal Z\), \[\|[\theta_j(z),d_{s,j}]\|\leq \frac{2\|z\|}{j}.\] The coordinatewise homomorphisms therefore induce a unital \(*\)-homomorphism \(z\mapsto[(\theta_j(z))_j]\) into \(A^\omega\cap D'\). It is injective because \(\mathcal Z\) is simple and the unit of \(A^\omega\) is nonzero. This proves Equation (7).

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 (7) to obtain \(\iota_i(\mathcal Z)\subseteq D_{i-1}'\), define \(C_i\) as in the statement, and put \[D_i=C^*(D_{i-1},C_i).\] For \(b\in\iota_i(\mathcal Z)\), \(d\in D_{i-1}\), 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 \(\operatorname{T}(C)\) is nonempty.

The trace \(\rho\) is faithful by simplicity of \(\mathcal Z\). Choose a non-scalar self-adjoint element \(c\in\mathcal Z\). Then \(\rho((c-\rho(c)1)^2)>0\), so rescaling \(c-\rho(c)1\) gives a self-adjoint \(b\in\mathcal Z\) with \(\rho(b)=0\) and \(\rho(b^2)=1\). Set \[x_{i,g}=\bar\alpha_g(\iota_i(b)),\qquad M=\|b\|.\] Every \(\varphi\in\operatorname{T}(C)\) restricts along each translated unital embedding of \(\mathcal Z\) to \(\rho\). This proves Equation (5), and Equation (6) follows from the action law. ◻

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 \[Z(\mathcal M_i)\subseteq Z(\mathcal M)=\mathbb C1.\] 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. ◻

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 (5). 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 \[ \varphi(z_{N,g_1}\cdots z_{N,g_d}) =N^{-d/2} \sum_{i_1,\ldots,i_d=1}^N \varphi(x_{i_1,g_1}\cdots x_{i_d,g_d}). \tag{12}\] 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 5 then factors the expectation over the distinct rows. A singleton block gives zero by Equation (5). 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 (12) 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 (10), 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 \[ k_{N_j;g,h}-\varphi_j(k_{N_j;g,h})1 =\frac1{N_j}\sum_{i=1}^{N_j}y_{j,i;g,h}. \tag{13}\]

Let \(U_j\) be an ordered word having \(d\) fluctuation factors and \(r\geq1\) factors of the form on the left of Equation (13). 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 5. 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 \[ \left|\varphi_j(U_j)\right| \leq C_{d,r,M}\, N_j^{\lfloor(d+r)/2\rfloor-d/2-r} \leq C_{d,r,M}\,N_j^{-r/2} \longrightarrow0, \tag{14}\] 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 (13). Every term containing a centered covariance factor tends to zero by Equation (14). The remaining terms converge by the fluctuation calculation and Equation (10). This proves convergence of every ordered word. Since all generators are self-adjoint, linearity also gives the \(*\)-polynomial formulation in Equation (11). ◻

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 \[\operatorname{Tr}\bigl(u(a^2-b^2)\bigr) =\operatorname{Tr}\bigl(|d|(a+b)\bigr).\] In a basis diagonalizing \(d\), positivity gives \(a_{jj}+b_{jj}\geq|a_{jj}-b_{jj}|=|d_j|\). Thus \[\operatorname{Tr}\bigl(|d|(a+b)\bigr) =\sum_j |d_j|(a_{jj}+b_{jj}) \geq\sum_j d_j^2=\operatorname{Tr}(d^2).\] On the other hand, \(u\) is a contraction, so trace-norm duality gives \[\operatorname{Tr}\bigl(u(a^2-b^2)\bigr)\leq\operatorname{Tr}|a^2-b^2|.\] These two inequalities prove the assertion. ◻

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 \[\bigl|\mathbb E\exp(\mathrm{i}s q_B)\bigr| =\prod_{j=1}^d(1+4s^2\lambda_j^2)^{-1/4}.\] 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 (17).

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, \[\int_{\mathbb R}\cosh(cx)\,d\nu(x) =\sum_{r=0}^{\infty} \frac{c^{2r}}{(2r)!}\mathbb E(q_B^{2r}) =\mathbb E\cosh(cq_B)<\infty.\] 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 \(\{z\in\mathbb C:|\operatorname{Im}z|<c\}\). 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 \[ \int_{|x|>K}|x|^d\,d\mu_j(x) \leq C_r K^{d-2r} \qquad(K>0). \tag{18}\]

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 (18) shows that \[\int x^d\,d\mu(x)=\mathbb E(q_B^d).\] This holds for every \(d\). Lemma 9 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 7. Step 1. Choice of parameters. Fix \(H,\varepsilon,m\) as in the statement. Choose \(t>0\) so that \[ (1+t^2)^{-1/4}<\varepsilon/4. \tag{19}\] Next choose \(\delta>0\) and \(0<\eta<1/2\) so that \[ t\sqrt{2}\,(\sqrt{\delta}+2\eta)<\varepsilon/4. \tag{20}\] 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 \[ p(s)=\sum_{r=1}^{D}c_rs^r, \qquad \sup_{0\leq s\leq1}|p(s)-\sqrt{s}|<\eta/\sqrt n. \tag{21}\] 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 \[P_E=\frac1n\sum_{g\in E}v_gv_g^{\mathsf T}, \qquad B_E=p(P_E), \qquad q_E=\langle W,B_EW\rangle-\operatorname{Tr}(B_E).\] 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 (21) therefore gives \[ \|B_E-P_E^{1/2}\|_{\mathrm{HS}}^2 \leq \operatorname{rank}(P_E)\,\eta^2/n \leq\eta^2, \qquad \|P_E^{1/2}\|_{\mathrm{HS}}^2=\operatorname{Tr}(P_E)=1. \tag{22}\] In particular, \[ 1-\eta\leq\|B_E\|_{\mathrm{HS}}\leq1+\eta. \tag{23}\]

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 8 implies \[\|P_F^{1/2}-P_{hF}^{1/2}\|_{\mathrm{HS}}^2 \leq\|P_F-P_{hF}\|_1 \leq\frac{|F\mathbin{\triangle}hF|}{n}<\delta.\] Combining this with Equation (22) yields \[\|B_F-B_{hF}\|_{\mathrm{HS}}\leq\sqrt{\delta}+2\eta.\] 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 9 gives \[ \|q_{hF}-q_F\|_{L^2} \leq\sqrt{2}\,(\sqrt{\delta}+2\eta). \tag{24}\] Also, Equation (23) and \(\eta<1/2\) give, for every integer \(\ell\geq1\), \[ \bigl|\mathbb E\exp(\mathrm{i}\ell t q_F)\bigr| \leq \bigl(1+4\ell^2t^2\|B_F\|_{\mathrm{HS}}^2\bigr)^{-1/4} \leq(1+t^2)^{-1/4}. \tag{25}\] 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 \[P_E^r =\frac1{n^r} \sum_{g_1,\ldots,g_r\in E} R_{g_1,g_2}\cdots R_{g_{r-1},g_r} v_{g_1}v_{g_r}^{\mathsf T}.\] Taking its quadratic form and its trace, respectively, shows that \[ q_E= \sum_{r=1}^{D}\frac{c_r}{n^r} \sum_{g_1,\ldots,g_r\in E} R_{g_1,g_2}\cdots R_{g_{r-1},g_r} \bigl(\xi_{g_1}\xi_{g_r}-R_{g_r,g_1}\bigr). \tag{26}\] The product of covariance entries is interpreted as \(1\) when \(r=1\).

Use the same prescription in \(C\) by setting \[\begin{align*} \widetilde Q_{N,E} &= \sum_{r=1}^{D}\frac{c_r}{n^r} \sum_{g_1,\ldots,g_r\in E} k_{N;g_1,g_2}\cdots k_{N;g_{r-1},g_r} \bigl(z_{N,g_1}z_{N,g_r}-k_{N;g_r,g_1}\bigr), \tag{27}\\ Q_{N,E} &=\tfrac12\bigl(\widetilde Q_{N,E} +\widetilde Q_{N,E}^{\,*}\bigr). \tag{28}\end{align*}\] Multiplication in Equation (27) 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 (9) 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 \[ Q_{N,hF}=\bar\alpha_h(Q_{N,F}) \qquad(h\in H,\ N\geq1). \tag{29}\]

Step 4. Transfer of moments and scalar laws. Consider now any sequence \(N_j\to\infty\) and extreme traces \(\varphi_j\in\operatorname{T}(C)\) such that \[\varphi_j(k_{N_j;g,g'})\longrightarrow R_{g,g'} \qquad(g,g'\in J).\] Proposition 6 gives convergence of every ordered mixed moment of the empirical variables. Any moment of the polynomials in Equations (27)–(28) is a finite linear combination of such moments. Since the limiting variables commute and Equation (26) is real-valued, it follows that the \(Q_{N_j,E}\) converge jointly in moments under \(\varphi_j\) to the \(q_E\). In particular, \[ \|Q_{N_j,hF}-Q_{N_j,F}\|_{2,\varphi_j} \longrightarrow \|q_{hF}-q_F\|_{L^2}. \tag{30}\] Lemma 10 also gives \[ \varphi_j\bigl(\exp(\mathrm{i}s Q_{N_j,F})\bigr) \longrightarrow\mathbb E\exp(\mathrm{i}s q_F) \qquad(s\in\mathbb R). \tag{31}\] 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 (18), is sufficient for Equation (31).

Step 5. Uniformity over all traces. Set \(V_N=\exp(\mathrm{i}tQ_{N,F})\). For any self-adjoint \(X,Y\) in a unital \(\mathrm{C}^*\)-algebra and any tracial state \(\varphi\), differentiation gives \[\frac{d}{ds} \bigl(e^{\mathrm{i}(1-s)tX}e^{\mathrm{i}stY}\bigr) =e^{\mathrm{i}(1-s)tX}\,\mathrm{i}t(Y-X)\,e^{\mathrm{i}stY}.\] The tracial \(2\)-seminorm is unchanged by left and right unitary multiplication. Integration and the triangle inequality therefore give \[ \|e^{\mathrm{i}tX}-e^{\mathrm{i}tY}\|_{2,\varphi} \leq t\|X-Y\|_{2,\varphi}. \tag{32}\] In view of Equations (29), (30), and (24), every sequence of traces and indices with convergent covariance as above satisfies \[\begin{align*} \limsup_{j\to\infty} \|\bar\alpha_h(V_{N_j})-V_{N_j}\|_{2,\varphi_j} &\leq t\sqrt{2}\,(\sqrt{\delta}+2\eta)<\varepsilon/4, \tag{33}\\ \lim_{j\to\infty}|\varphi_j(V_{N_j}^{\ell})| &= \bigl|\mathbb E\exp(\mathrm{i}\ell t q_F)\bigr| \leq(1+t^2)^{-1/4}<\varepsilon/4. \tag{34}\end{align*}\] Here \(h\in H\) and \(1\leq\ell\leq m\), and the strict inequalities are Equations (20) and (19).

It follows that, for all sufficiently large \(N\), every extreme trace \(\varphi\in\operatorname{T}(C)\) satisfies \[ \|\bar\alpha_h(V_N)-V_N\|_{2,\varphi}<\varepsilon/2 \quad(h\in H),\qquad |\varphi(V_N^\ell)|<\varepsilon/2 \quad(1\leq\ell\leq m). \tag{35}\] 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 (33) or Equation (34) contradicts the failure at tolerance \(\varepsilon/2\). 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 (35) 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 \(\operatorname{T}(C)\). 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 \(\varepsilon^2/4\) and \(\varepsilon/2\), respectively, are closed and convex and contain all extreme traces. By the Krein–Milman Theorem they contain \(\operatorname{T}(C)\). Thus the suprema in Equations (15) and (16) are at most \(\varepsilon/2\), and in particular strictly less than \(\varepsilon\). 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 \(\sigma_j\in\operatorname{T}(A)\) 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. ◻