Proof of Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and $L^2$ Approximation

Each result cited is universally quantified over the data in its own statement.

Throughout, $I$ alone denotes the identity map of the Hilbert space in play, while the cost of a coupling $\gamma$ is always written with its argument, $I(\gamma)$; the letter $M$ always denotes the algebra of operators, and the second moment $M(\lambda)$ does not occur in this proof.

Step 0 (Standing facts). By Tracial W*-Probability Spaces §space, $(H,M,\Omega)$ is a cyclic tracial operator algebra. Hence $M\subseteq\mathcal{L}(H)$ by Cyclic Tracial Operator Algebras and Their Traces §triple; $I\in M$ and $M$ is closed under sums, complex multiples, composition and adjoints by Cyclic Tracial Operator Algebras and Their Traces §star-algebra; $\lVert\Omega\rVert=1$ by Cyclic Tracial Operator Algebras and Their Traces §cyclic; and $\langle\Omega,ST\Omega\rangle=\langle\Omega,TS\Omega\rangle$ for all $S,T\in M$ by Cyclic Tracial Operator Algebras and Their Traces §tracial. The same facts hold for any other tracial W*-probability space, such as $(K,N,\Psi)$ in Clause 5.

(a) Values lie in $M$. Let $r\in\mathbb{N}$ and let $T$ be an $r$-tuple of elements of $M$. Apply Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §transport with $K=H$, with $\mathcal{A}=M$, which contains $I,T_{1},\dots,T_{r}$ and has the required closure properties by the above, and with $\Phi$ the inclusion map $\Phi(A)=A$, which satisfies the four required identities trivially. It gives $p(T)\in M$ for every $p\in\mathcal{P}_{r}$; in particular $T_{w}=x_{w}(T)\in M$ for every $w\in W_{r}$, the equality being Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §evaluation.

(b) Self-adjointness. By Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus, a linear map $A:H\to H$ is self-adjoint if and only if $A$ is an adjoint of $A$, so a self-adjoint $A$ satisfies $A^{*}=A$ by the uniqueness of adjoints in Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-unique. The same calculus clause says that $I$ is its own adjoint and that $S+T$ and $cT$ have the adjoints $S^{*}+T^{*}$ and $\overline{c}\,T^{*}$. Consequently, if $S_{1},\dots,S_{q}$ are self-adjoint elements of $\mathcal{L}(H)$ and $a_{0},a_{1},\dots,a_{q}$ are real, then by induction on $q$ the operator $a_{0}I+\sum_{l=1}^{q}a_{l}S_{l}$ has itself as adjoint, because $\overline{a}=a$ for real $a$; hence it is self-adjoint.

(c) Self-adjoint vectors. $J\Omega=\Omega$ by The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §conjugation, so $\Omega\in H_{\mathrm{sa}}$ by Conjugation of a Complex Hilbert Space §fixed. By Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §self-adjoint, $S\Omega\in H_{\mathrm{sa}}$ for every self-adjoint $S\in M$, and $\langle\xi,\eta\rangle$ is real for all $\xi,\eta\in H_{\mathrm{sa}}$.

Clause 1. Let $R>0$ be real with $\lVert s_{j}\rVert_{\mathrm{op}}\le R$ for every $j\in[d]$. Each $s_{j}$ is a self-adjoint element of $\mathcal{L}(H)$ by Step 0, and $\lVert\Omega\rVert=1$. For $u,v\in W_{d}$ the operators $s_{u},s_{v}$ lie in $M$ by Step 0(a), so $\langle\Omega,s_{u}s_{v}\Omega\rangle=\langle\Omega,s_{v}s_{u}\Omega\rangle$ by the tracial property in Step 0. Therefore The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State §law, applied with $n=d$ and $T=s$, shows that the map $p\mapsto\langle\Omega,p(s)\Omega\rangle$ lies in $\Sigma_{d,R}$; by Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law this map is $\lambda_{s}$. Such an $R$ exists: each $\lVert s_{j}\rVert_{\mathrm{op}}$ is the infimum of a set of bounds bounded below by $0$, hence nonnegative by Bounded Linear Maps between Complex Normed Spaces and the Operator Norm §operator-norm, so $R_{0}=1+\sum_{j=1}^{d}\lVert s_{j}\rVert_{\mathrm{op}}$ satisfies $R_{0}\ge1>0$ and $\lVert s_{j}\rVert_{\mathrm{op}}\le R_{0}$ for every $j$. Thus $\lambda_{s}\in\Sigma_{d,R_{0}}\subseteq\Sigma_{d}$, the inclusion by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law. Finally $s_{j}\Omega\in H_{\mathrm{sa}}$ by Step 0(c).

Clause 2. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, $x_{i}(s)=s_{i}$, so $\lambda_{s}(x_{i})=\langle\Omega,s_{i}\Omega\rangle$. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §homomorphism and the same values clause, $(x_{i}x_{j})(s)=s_{i}s_{j}$, so $\lambda_{s}(x_{i}x_{j})=\langle\Omega,s_{i}s_{j}\Omega\rangle$, and this equals $\langle s_{i}\Omega,s_{j}\Omega\rangle$ by Self-Adjoint Operator applied to $s_{i}$ with the vectors $\Omega$ and $s_{j}\Omega$. Since $\Omega$, $s_{i}\Omega$ and $s_{j}\Omega$ lie in $H_{\mathrm{sa}}$ (Step 0(c) and Clause 1), both numbers are real by Step 0(c).

Clause 3. Since $c_{i}$ and $A_{ij}$ are real, hence complex, numbers, $u_{i}\in M$ by the closure properties in Step 0, and $u_{i}$ is self-adjoint by Step 0(b); so $u$ is a self-adjoint $n$-tuple in $M$. Sums and scalar multiples of operators are formed pointwise by Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §maps, which gives $u_{i}\Omega=c_{i}\Omega+\sum_{j=1}^{d}A_{ij}s_{j}\Omega$. Let $a^{T}$ be the tuple of $T$ in Affine Data and Affine Substitutions of Noncommutative Polynomials §tuple, so that $a^{T}_{i}=c_{i}1+\sum_{j=1}^{d}A_{ij}x_{j}$. Evaluation at $s$ is linear by Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §evaluation, and $1(s)=I$ and $x_{j}(s)=s_{j}$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values; hence

that is, the $n$-tuple $a^{T}(s)=(a^{T}_{1}(s),\dots,a^{T}_{n}(s))$ equals $u$. By Affine Data and Affine Substitutions of Noncommutative Polynomials §substitution, $\sigma_{T}=\sigma_{a^{T}}$, and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution, applied with $m=d$, the $n$-tuple $a^{T}$ in $\mathcal{P}_{d}$ and the $d$-tuple $s$, gives $(\sigma_{T}(p))(s)=p(u)$ for every $p\in\mathcal{P}_{n}$. Therefore, by Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law applied to $s$ and to $u$,

so $\lambda_{u}=\lambda_{s}\circ\sigma_{T}$.

Clause 4. Write $r=(s,t)$, so $r_{j}=s_{j}$ and $r_{d+j}=t_{j}$ for $j\in[d]$; each entry is a self-adjoint element of $M$, so $r$ is a self-adjoint $2d$-tuple in $M$. Clause 1 was proved above for an arbitrary number of variables; applied to the $2d$-tuple $r$ it gives $\lambda_{r}\in\Sigma_{2d}$, so $\lambda_{r}$ is a tracial state on $\mathcal{P}_{2d}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law; applied to $s$ and $t$ it gives $\lambda_{s},\lambda_{t}\in\Sigma_{d}$.

Marginals. By Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals, $\iota^{1}=\sigma_{a}$ and $\iota^{2}=\sigma_{b}$ for the $d$-tuples $a=(x_{1},\dots,x_{d})$ and $b=(x_{d+1},\dots,x_{2d})$ in $\mathcal{P}_{2d}$. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, $x_{j}(r)=s_{j}$ and $x_{d+j}(r)=t_{j}$ for $j\in[d]$, so $a(r)=s$ and $b(r)=t$. Hence Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution, with $m=2d$, $n=d$ and the $2d$-tuple $r$, gives $(\iota^{1}(p))(r)=p(s)$ and $(\iota^{2}(p))(r)=p(t)$ for every $p\in\mathcal{P}_{d}$, and therefore $\lambda_{r}(\iota^{1}(p))=\lambda_{s}(p)$ and $\lambda_{r}(\iota^{2}(p))=\lambda_{t}(p)$. By Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling, $\lambda_{r}\in\Pi(\lambda_{s},\lambda_{t})$.

Cost. By Couplings of Two Noncommutative Laws and Their Quadratic Cost §cost, the cost of $\lambda_{r}$ is $I(\lambda_{r})=\lambda_{r}(\Delta_{d})$ with $\Delta_{d}=\sum_{j=1}^{d}(x_{j}-x_{d+j})(x_{j}-x_{d+j})$. For $j\in[d]$ let $e_{j}=s_{j}+(-1)t_{j}\in M$, which is self-adjoint by Step 0(b), so $e_{j}^{*}=e_{j}$. Linearity of evaluation, Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §homomorphism and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values give $\Delta_{d}(r)=\sum_{j=1}^{d}e_{j}e_{j}$, and linearity of the inner product in its second argument (Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces) gives $I(\lambda_{r})=\sum_{j=1}^{d}\langle\Omega,e_{j}e_{j}\Omega\rangle$. The last identity of Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus, with $T=e_{j}=e_{j}^{*}$ and $v=\Omega$, gives $\langle\Omega,e_{j}e_{j}\Omega\rangle=\lVert e_{j}\Omega\rVert^{2}$, and $e_{j}\Omega=s_{j}\Omega-t_{j}\Omega$ pointwise. Hence $I(\lambda_{r})=\sum_{j=1}^{d}\lVert s_{j}\Omega-t_{j}\Omega\rVert^{2}$.

Comparison. By The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §distance, $W_{2}(\lambda_{s},\lambda_{t})$ is the nonnegative square root of the infimum $\beta$ of the set $\{I(\gamma):\gamma\in\Pi(\lambda_{s},\lambda_{t})\}$, so $W_{2}(\lambda_{s},\lambda_{t})^{2}=\beta$. Since $\lambda_{r}$ belongs to $\Pi(\lambda_{s},\lambda_{t})$ and $\beta$ is a lower bound of that set, $W_{2}(\lambda_{s},\lambda_{t})^{2}\le I(\lambda_{r})$.

Clause 5. By Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §embedding, $\pi$ maps $M$ into $N$, so $\pi(s_{j})\in N$; and $\pi(s_{j})$ has the adjoint $\pi(s_{j}^{*})=\pi(s_{j})$, using $s_{j}^{*}=s_{j}$ (Step 0(b)). Hence $\pi(s_{j})$ is self-adjoint by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint-calculus, and $\pi(s)$ is a self-adjoint $d$-tuple in $N$. The identity $\pi(s_{j})\Psi=V_{\pi}s_{j}\Omega$ is the defining property of $V_{\pi}$ in Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §isometry, with $S=s_{j}$.

For the law, apply Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §transport with the $d$-tuple $s$, with $\mathcal{A}=M$ (Step 0), and with $\Phi=\pi$, regarded as a map into $\mathcal{L}(K)$ because $N\subseteq\mathcal{L}(K)$ (Step 0 for $(K,N,\Psi)$); $\pi$ satisfies the four required identities by Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §embedding. It gives $p(s)\in M$ and $\pi(p(s))=p(\pi(s))$ for every $p\in\mathcal{P}_{d}$. Let $\tau_{N}$ and $\tau_{M}$ be the traces of Tracial W*-Probability Spaces §trace, given by $\tau_{N}(S)=\langle\Psi,S\Psi\rangle$ and $\tau_{M}(S)=\langle\Omega,S\Omega\rangle$ by Cyclic Tracial Operator Algebras and Their Traces §trace. Using the trace identity $\tau_{N}(\pi(S))=\tau_{M}(S)$ of Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §embedding with $S=p(s)$,

for every $p\in\mathcal{P}_{d}$, the outer equalities by Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law. Hence $\lambda_{\pi(s)}=\lambda_{s}$.

Clause 6. Let $\iota_{\mathbb{R}}:\mathbb{N}\to\mathbb{R}$ be the canonical map of The Canonical Map from the Natural Numbers to a Field; it satisfies $\iota_{\mathbb{R}}(k+1)=\iota_{\mathbb{R}}(k)+1$ and $\iota_{\mathbb{R}}(1)=1$, so $1\le\iota_{\mathbb{R}}(l)\le\iota_{\mathbb{R}}(k)$ whenever $l\le k$. For $k\in\mathbb{N}$ put $\varepsilon_{k}=\iota_{\mathbb{R}}(k)^{-1}$; by Order Reversal under Reciprocals, and Summability of the Reciprocals of the Squares §reciprocal, $\varepsilon_{k}$ exists, $\varepsilon_{k}>0$, and $\varepsilon_{k}\le\varepsilon_{l}$ whenever $l\le k$. Convergence in $H$ refers to the metric $(\xi,\eta)\mapsto\lVert\xi-\eta\rVert$ of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces, which is a metric by claim 3 of The Induced Norm is a Norm, and Induces a Metric.

Step 6.1 (Existence). Let $\mathcal{S}$ be the set of $d$-tuples of elements of $M$, and for $k\in\mathbb{N}$ let $A_{k}\subseteq\mathcal{S}$ be the set of self-adjoint $d$-tuples $s$ in $M$ with $\lVert X_{j}-s_{j}\Omega\rVert<\varepsilon_{k}$ for every $j\in[d]$. Each $A_{k}$ is nonempty: for each $j\in[d]$, Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §self-adjoint, applied to $X_{j}\in H_{\mathrm{sa}}$ and $\varepsilon_{k}$, provides a self-adjoint $S\in M$ with $\lVert X_{j}-S\Omega\rVert<\varepsilon_{k}$, and making these finitely many choices, one for each $j\in[d]$, yields an element of $A_{k}$. By Axiom of Countable Choice there is a sequence $(s^{k})_{k\in\mathbb{N}}$ with $s^{k}\in A_{k}$ for every $k$. Given real $\varepsilon>0$, claim 3 of The Archimedean Property of the Real Numbers gives $N\in\mathbb{N}$ with $\varepsilon_{N}<\varepsilon$, and then $\lVert s^{k}_{j}\Omega-X_{j}\rVert<\varepsilon_{k}\le\varepsilon_{N}<\varepsilon$ for all $k\ge N$ and $j\in[d]$, using the symmetry of the metric. So $(s^{k}_{j}\Omega)_{k\in\mathbb{N}}$ converges to $X_{j}$ for every $j\in[d]$, by Convergent Sequence in a Metric Space.

Step 6.2 (The basic estimate). Let $s$ and $t$ be self-adjoint $d$-tuples in $M$. Then $\lambda_{s},\lambda_{t}\in\Sigma_{d}$ by Clause 1. By Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §laws, $(\Sigma^{2}_{d},\widehat{W}_{2})$ is the metric completion of $(\Sigma_{d},W_{2})$ with canonical map $\kappa_{d}$, so The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §isometry and Clause 4 give

Consequently, if $\varepsilon>0$ is real and $\lVert s_{j}\Omega-t_{j}\Omega\rVert<\varepsilon/d$ for every $j\in[d]$, then $\widehat{W}_{2}(\kappa_{d}(\lambda_{s}),\kappa_{d}(\lambda_{t}))<\varepsilon$. Indeed, claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field, applied to the nonnegative numbers $\lVert s_{j}\Omega-t_{j}\Omega\rVert$ and $\varepsilon/d$, gives $\lVert s_{j}\Omega-t_{j}\Omega\rVert^{2}<\varepsilon^{2}/d^{2}$, so the right side of (E) is less than $d\cdot\varepsilon^{2}/d^{2}=\varepsilon^{2}/d\le\varepsilon^{2}$, since $d\ge1$ by Natural Numbers. As $\widehat{W}_{2}$ is a metric by The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §metric, its values are nonnegative by condition 1 of Metric Space, and claim 1 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field again gives the assertion.

Step 6.3 (Convergence). Let $(s^{k})_{k\in\mathbb{N}}$ be any sequence of self-adjoint $d$-tuples in $M$ with $(s^{k}_{j}\Omega)_{k}$ converging to $X_{j}$ for every $j\in[d]$, and put $\mu_{k}=\kappa_{d}(\lambda_{s^{k}})$. Given real $\varepsilon>0$, choose for each $j\in[d]$ some $N_{j}\in\mathbb{N}$ with $\lVert s^{k}_{j}\Omega-X_{j}\rVert<\varepsilon/(2d)$ for $k\ge N_{j}$, and let $N$ be the largest of the finitely many $N_{j}$. For $k,l\ge N$ the triangle inequality of claim 3 of The Induced Norm is a Norm, and Induces a Metric gives $\lVert s^{k}_{j}\Omega-s^{l}_{j}\Omega\rVert<\varepsilon/d$ for every $j$, so $\widehat{W}_{2}(\mu_{k},\mu_{l})<\varepsilon$ by Step 6.2. Thus $(\mu_{k})$ is a Cauchy sequence in the sense of Cauchy Sequence in a Metric Space. The space $(\Sigma^{2}_{d},\widehat{W}_{2})$ is complete by The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §complete, so $(\mu_{k})$ converges by Complete Metric Space.

Step 6.4 (Independence). Let $(s^{k})$ and $(t^{k})$ be two sequences as in Step 6.3, and let $\mu$ and $\nu$ be the limits of $\mu_{k}=\kappa_{d}(\lambda_{s^{k}})$ and $\nu_{k}=\kappa_{d}(\lambda_{t^{k}})$. Given real $\varepsilon>0$, choose $N\in\mathbb{N}$ (the largest of finitely many thresholds) such that, for all $k\ge N$ and $j\in[d]$, $\lVert s^{k}_{j}\Omega-X_{j}\rVert<\varepsilon/(4d)$, $\lVert t^{k}_{j}\Omega-X_{j}\rVert<\varepsilon/(4d)$ and $\widehat{W}_{2}(\mu_{k},\mu)<\varepsilon/2$. For $k\ge N$ the triangle inequality in $H$ gives $\lVert s^{k}_{j}\Omega-t^{k}_{j}\Omega\rVert<\varepsilon/(2d)$, so $\widehat{W}_{2}(\nu_{k},\mu_{k})<\varepsilon/2$ by Step 6.2, and the triangle inequality and symmetry of the metric $\widehat{W}_{2}$ (conditions 3 and 4 of Metric Space) give $\widehat{W}_{2}(\nu_{k},\mu)<\varepsilon$. Hence $(\nu_{k})$ converges to $\mu$ as well as to $\nu$, and $\mu=\nu$ by Uniqueness of Limits in a Metric Space. $\blacksquare$