Proof of Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs

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

Notation. $M$ and $M_{k}$ name sets of operators, the algebras of tracial W*-probability spaces; the second moment $M(\lambda)$ of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws is not used. As in Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §dimensions, $I(\gamma)$ is the cost of a coupling $\gamma$. For $k\in\mathbb{N}$ let $\theta_{k}=(\tfrac{1}{2})^{k}$, the natural power of The Real Numbers: Standing Notation and Background §numbers; thus $\theta_{k}>0$ and $\theta_{k+1}=\tfrac{1}{2}\theta_{k}$, and $(\theta_{k})_{k\in\mathbb{N}}$ converges to $0$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §geometric with $r=\tfrac{1}{2}$. For a tracial W*-probability space, $H_{\mathrm{sa}}$ is the set of fixed vectors of its conjugation, as in Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws. Every variable $x_{i}$ is self-adjoint: its support is the one-letter word $(i)$, which is its own reversal, so $x_{i}^{*}=x_{i}$ by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §adjoint.

Proof of clause 1 (Laws). Let $\mu\in\Sigma^{2}_{d}$. Choices are made in this order: first a representing Cauchy sequence $y$ (Step 1, a single instance), then the sequence $(\lambda_{k})$ by countable choice (Step 1), then the sequence $(\gamma_{k})$ by countable choice, given $(\lambda_{k})$ (Step 2). Every later object is either a single instance of an existence statement (Steps 3 and 5) or determined by the preceding data.

Step 1. Approximating bounded laws. By Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §laws and The Metric Completion of a Metric Space §completion, $\mu=[y]$ for some Cauchy sequence $y=(y_{n})_{n\in\mathbb{N}}$ in $(\Sigma_{d},W_{2})$, and 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 §density the sequence $(\kappa_{d}(y_{n}))_{n\in\mathbb{N}}$ converges to $\mu$ in $(\Sigma^{2}_{d},\widehat{W}_{2})$. For $k\in\mathbb{N}$ let

Each $A_{k}$ is nonempty: Convergent Sequence in a Metric Space with the real number $\theta_{k}>0$ gives $K\in\mathbb{N}$ with $y_{K}\in A_{k}$. By Axiom of Countable Choice (with $S=\Sigma_{d}$) there is a sequence $(\lambda_{k})_{k\in\mathbb{N}}$ with $\lambda_{k}\in A_{k}$ for every $k$. Since $\widehat{W}_{2}$ 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, conditions 3 and 4 of Metric Space and 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 give

Moreover $(\kappa_{d}(\lambda_{k}))_{k\in\mathbb{N}}$ converges to $\mu$: for real $\eta>0$ there is $K$ with $\theta_{k}<\eta$ for all $k\ge K$, and then $\widehat{W}_{2}(\kappa_{d}(\lambda_{k}),\mu)<\theta_{k}<\eta$. $\qquad\text{(2)}$

Step 2. Optimal couplings. For $k\in\mathbb{N}$ let $B_{k}$ be the set of $\gamma\in\Pi(\lambda_{k},\lambda_{k+1})$ with $I(\gamma)=W_{2}(\lambda_{k},\lambda_{k+1})^{2}$. Each $B_{k}$ is nonempty: by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law there are reals $r,r'>0$ with $\lambda_{k}\in\Sigma_{d,r}$ and $\lambda_{k+1}\in\Sigma_{d,r'}$; both lie in $\Sigma_{d,R}$ for $R=\max(r,r')$ by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §monotone; so The Noncommutative Wasserstein Distance: Existence of Optimal Couplings, Symmetry, Separation, a Moment Bound, Weak-Star Lower Semicontinuity, and Displacement Interpolation §attained gives an optimal coupling, which lies in $B_{k}$ by The Noncommutative Quadratic Wasserstein Distance and Optimal Couplings §optimal. By Axiom of Countable Choice (with $S$ the set of maps $\mathcal{P}_{2d}\to\mathbb{C}$) there is a sequence $(\gamma_{k})_{k\in\mathbb{N}}$ with $\gamma_{k}\in B_{k}$ for every $k$. Squaring the nonnegative numbers in (1),

Step 3. Consistent joint laws. Apply Gluing a Chain of Noncommutative Couplings into Consistent Joint Laws to $(\lambda_{k})$ and $(\gamma_{k})$, and fix a sequence $(\Gamma_{k})_{k\in\mathbb{N}}$ with $\Gamma_{k}\in\Sigma_{kd}$ and the properties Gluing a Chain of Noncommutative Couplings into Consistent Joint Laws §consistent, Gluing a Chain of Noncommutative Couplings into Consistent Joint Laws §links and Gluing a Chain of Noncommutative Couplings into Consistent Joint Laws §blocks, with the substitutions $\beta_{k}$, $\delta_{k}$ and $\varepsilon_{k}$ of that lemma.

Step 4. The spaces and the embeddings. For $k\in\mathbb{N}$ let $H_{k}=\mathcal{H}_{\Gamma_{k}}$, $M_{k}=\mathcal{M}_{\Gamma_{k}}$ and $\Omega_{k}=\Omega_{\Gamma_{k}}$, and write $L^{(k)}_{p}$ for the left multiplication by $p\in\mathcal{P}_{kd}$ on $H_{k}$ (Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §multiplication). By The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star, $(H_{k},M_{k},\Omega_{k})$ is a tracial W*-probability space with trace $\tau_{\Gamma_{k}}$. Apply Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation with $\gamma=\Gamma_{k+1}\in\Sigma_{(k+1)d}$, $m=(k+1)d$, $n=kd$ and the $kd$-tuple $(x_{1},\dots,x_{kd})$ of self-adjoint polynomials; its substitution is $\beta_{k}$, so the marginal law of that lemma is $\Gamma_{k+1}\circ\beta_{k}=\Gamma_{k}$ by Gluing a Chain of Noncommutative Couplings into Consistent Joint Laws §consistent. Let $\pi_{k}:M_{k}\to M_{k+1}$ be the embedding of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §embedding. By Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §homomorphism, $\pi_{k}$ is linear, unital, multiplicative and preserves adjoints, and $\tau_{\Gamma_{k+1}}(\pi_{k}(S))=\tau_{\Gamma_{k}}(S)$ for $S\in M_{k}$; so $\pi_{k}$ is a trace-preserving embedding of $(H_{k},M_{k},\Omega_{k})$ into $(H_{k+1},M_{k+1},\Omega_{k+1})$ in the sense of Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §embedding. The same clause and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values give

Step 5. The inductive limit. Apply Inductive Limits of Tracial W*-Probability Spaces along Trace-Preserving Embeddings to the spaces and embeddings of Step 4, and fix a tracial W*-probability space $(H,M,\Omega)$ and trace-preserving embeddings $\rho_{k}$ of $(H_{k},M_{k},\Omega_{k})$ into $(H,M,\Omega)$ with the property Inductive Limits of Tracial W*-Probability Spaces along Trace-Preserving Embeddings §compatible.

Step 6. The bounded tuples and their laws. Each $L^{(k)}_{x_{i}}$ ($i\in[kd]$) lies in $M_{k}$ by The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots §star-algebra and is self-adjoint by Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §adjoint. For $k\in\mathbb{N}$, with the convention of Gluing a Chain of Noncommutative Couplings into Consistent Joint Laws that $(k-1)d+j$ means $j$ when $k=1$ (used for every such index below), let $u^{k}$ be the $d$-tuple with $u^{k}_{j}=L^{(k)}_{x_{(k-1)d+j}}$ and let $s^{k}=\rho_{k}(u^{k})$, the $d$-tuple with $s^{k}_{j}=\rho_{k}(u^{k}_{j})$ ($j\in[d]$). Then $u^{k}$ is a self-adjoint $d$-tuple in $M_{k}$ (Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §tuple), and by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §embedding $s^{k}$ is a self-adjoint $d$-tuple in $M$ with $\lambda_{s^{k}}=\lambda_{u^{k}}$.

Law of $u^{k}$. Let $\ell^{k}=(L^{(k)}_{x_{1}},\dots,L^{(k)}_{x_{kd}})$ and let $e=(x_{(k-1)d+1},\dots,x_{kd})$ be the $d$-tuple of $\varepsilon_{k}$. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, $e(\ell^{k})=u^{k}$, so for $p\in\mathcal{P}_{d}$, Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution gives $p(u^{k})=(\varepsilon_{k}p)(\ell^{k})$, and The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §law for $\Gamma_{k}$ gives $\langle\Omega_{k},(\varepsilon_{k}p)(\ell^{k})\Omega_{k}\rangle=\Gamma_{k}(\varepsilon_{k}p)$. By Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law and Gluing a Chain of Noncommutative Couplings into Consistent Joint Laws §blocks, $\lambda_{u^{k}}(p)=\Gamma_{k}(\varepsilon_{k}p)=\lambda_{k}(p)$. Hence

Consecutive pairs. Let $v^{k}$ be the $2d$-tuple in $M_{k+1}$ with $v^{k}_{i}=L^{(k+1)}_{x_{(k-1)d+i}}$ ($i\in[2d]$), a self-adjoint $2d$-tuple in $M_{k+1}$ by the first paragraph of this step. For $j\in[d]$, $(k-1)d+j\in[kd]$, so Inductive Limits of Tracial W*-Probability Spaces along Trace-Preserving Embeddings §compatible and (4) give $s^{k}_{j}=\rho_{k+1}(\pi_{k}(u^{k}_{j}))=\rho_{k+1}(L^{(k+1)}_{x_{(k-1)d+j}})=\rho_{k+1}(v^{k}_{j})$; and $v^{k}_{d+j}=L^{(k+1)}_{x_{kd+j}}=u^{k+1}_{j}$, so $\rho_{k+1}(v^{k}_{d+j})=s^{k+1}_{j}$. Thus $\rho_{k+1}(v^{k})=(s^{k},s^{k+1})$. The computation of the previous paragraph, with $\Gamma_{k+1}$, $\ell^{k+1}$ and the $2d$-tuple $(x_{(k-1)d+1},\dots,x_{(k+1)d})$ of $\delta_{k}$ in place of $\Gamma_{k}$, $\ell^{k}$ and $e$, and with Gluing a Chain of Noncommutative Couplings into Consistent Joint Laws §links in place of Gluing a Chain of Noncommutative Couplings into Consistent Joint Laws §blocks, gives $\lambda_{v^{k}}(q)=\Gamma_{k+1}(\delta_{k}q)=\gamma_{k}(q)$ for $q\in\mathcal{P}_{2d}$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §embedding (with $2d$ in place of $d$), $\lambda_{(s^{k},s^{k+1})}=\lambda_{v^{k}}=\gamma_{k}$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling and (3),

Each summand is nonnegative, so each is at most the sum; and for nonnegative reals $a,b$, $a^{2}<b^{2}$ implies $a<b$. Hence

Step 7. The limit tuple. Fix $j\in[d]$ and $k\in\mathbb{N}$. We show by induction on $l\ge k+1$ that $\lVert s^{k}_{j}\Omega-s^{l}_{j}\Omega\rVert<4\theta_{k}-4\theta_{l}$. For $l=k+1$ this is (6), since $4\theta_{k}-4\theta_{k+1}=2\theta_{k}$. If it holds for $l$, then the triangle inequality of the metric of $H$ (claim 3 of The Induced Norm is a Norm, and Induces a Metric and condition 4 of Metric Space) and (6) give

Hence $\lVert s^{k}_{j}\Omega-s^{l}_{j}\Omega\rVert<4\theta_{k}$ whenever $l>k$, and the left side is $0$ when $l=k$. Given real $\eta>0$, take $K$ with $\theta_{k}<\eta/4$ for all $k\ge K$; for $k,l\ge K$, by symmetry of the metric we may assume $l\ge k$, and then $\lVert s^{k}_{j}\Omega-s^{l}_{j}\Omega\rVert<\eta$. So $(s^{k}_{j}\Omega)_{k\in\mathbb{N}}$ is a Cauchy sequence in $H$ (Cauchy Sequence in a Metric Space). $H$ is a complex Hilbert space by Tracial W*-Probability Spaces §space and Cyclic Tracial Operator Algebras and Their Traces §triple, hence complete by Complex Hilbert Space; so by Complete Metric Space the sequence converges in $H$ to some $X_{j}\in H$. Each $s^{k}_{j}\Omega$ lies in $H_{\mathrm{sa}}$ by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law, and $H_{\mathrm{sa}}$ contains the limits of its convergent sequences by Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §self-adjoint; so $X_{j}\in H_{\mathrm{sa}}$.

Thus $X=(X_{1},\dots,X_{d})$ is an $L^{2}$ $d$-tuple of $(H,M,\Omega)$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples), and $(s^{k})_{k\in\mathbb{N}}$ is a sequence of self-adjoint $d$-tuples in $M$ with $(s^{k}_{j}\Omega)_{k}$ converging to $X_{j}$ for every $j$. By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §law, $\mathrm{law}(X)$ is the limit of $(\kappa_{d}(\lambda_{s^{k}}))_{k\in\mathbb{N}}$, which is the sequence $(\kappa_{d}(\lambda_{k}))_{k\in\mathbb{N}}$ by (5). That sequence converges to $\mu$ by (2), so $\mathrm{law}(X)=\mu$ by Uniqueness of Limits in a Metric Space.

Proof of clause 2 (Couplings). Step 1. Existence. Let $\gamma\in\Sigma^{2}_{2d}$. Clause 1, with $2d$ in place of $d$, gives a tracial W*-probability space $(H,M,\Omega)$ and an $L^{2}$ $2d$-tuple $Z$ of it with $\mathrm{law}(Z)=\gamma$. Let $X=(Z_{1},\dots,Z_{d})$ and $Y=(Z_{d+1},\dots,Z_{2d})$; their entries lie in $H_{\mathrm{sa}}$, so they are $L^{2}$ $d$-tuples by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples, and $(X,Y)=Z$ by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations. Hence $\mathrm{law}(X,Y)=\gamma$.

Step 2. Marginals and cost. Now let $(H,M,\Omega)$ be any tracial W*-probability space and $X,Y$ any $L^{2}$ $d$-tuples of it with $\mathrm{law}(X,Y)=\gamma$. By Affine Data and Affine Substitutions of Noncommutative Polynomials §coordinate, $\mathrm{pr}^{1}=(P^{1},0)$ with $P^{1}_{ij}=1$ if $j=i$ and $0$ otherwise, so by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, for $i\in[d]$,

likewise $P^{2}_{ij}=1$ exactly when $j=d+i$, so $(\mathrm{pr}^{2}(X,Y))_{i}=(X,Y)_{d+i}=Y_{i}$. Thus $\mathrm{pr}^{1}(X,Y)=X$ and $\mathrm{pr}^{2}(X,Y)=Y$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, applied to the $L^{2}$ $2d$-tuple $(X,Y)$ with $2d$ in place of $d$ and $d$ in place of $n$,

By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §coupling, $\lVert X-Y\rVert_{2}^{2}=\mathcal{I}(\mathrm{law}(X,Y))=\mathcal{I}(\gamma)$.

Proof of clause 3 (Almost optimal pairs). Let $\mu,\nu\in\Sigma^{2}_{d}$ and let $\varepsilon>0$ be real. By The Distance between Square-Integrable Noncommutative Laws is the Infimum of the Cost over Their Couplings §approximate there is $\gamma\in\Pi^{2}(\mu,\nu)$ with $\mathcal{I}(\gamma)\le\widehat{W}_{2}(\mu,\nu)^{2}+\varepsilon$; by Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §couplings, $\gamma\in\Sigma^{2}_{2d}$, $\mathrm{pr}^{1}_{\#}\gamma=\mu$ and $\mathrm{pr}^{2}_{\#}\gamma=\nu$. Clause 2 gives a tracial W*-probability space $(H,M,\Omega)$ and $L^{2}$ $d$-tuples $X,Y$ of it with $\mathrm{law}(X,Y)=\gamma$, and then $\mathrm{law}(X)=\mathrm{pr}^{1}_{\#}\gamma=\mu$, $\mathrm{law}(Y)=\mathrm{pr}^{2}_{\#}\gamma=\nu$ and $\lVert X-Y\rVert_{2}^{2}=\mathcal{I}(\gamma)\le\widehat{W}_{2}(\mu,\nu)^{2}+\varepsilon$.