Proof of Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings

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

Throughout, the letter $M$ denotes the algebra of operators; only the second moment $\widehat{M}$ of $L^{2}$ laws occurs below, never $M(\lambda)$. The identity map of $H$ is $I$, and the cost of $L^{2}$ laws is written $\mathcal{I}$.

Step 0 (Limits). (a) A sequence $(y_{k})$ in a metric space $(Z,e)$ converges to $y$ in the sense of Convergent Sequence in a Metric Space if and only if the real sequence $(e(y_{k},y))_{k}$ converges to $0$ in the sense of Limit of a Sequence of Real Numbers, since $|e(y_{k},y)-0|=e(y_{k},y)\ge0$ by condition 1 of Metric Space. Likewise a real sequence converges to $L$ in the sense of Limit of a Sequence of Real Numbers if and only if it converges to $L$ in the real line of The Absolute Value Metric on the Real Line; so limits of real sequences are unique by Uniqueness of Limits in a Metric Space.

(b) Distances. If $y_{k}\to y$ and $z_{k}\to z$ in a metric space $(Z,e)$, then $e(y_{k},z_{k})\to e(y,z)$. Indeed, conditions 3 and 4 of Metric Space give $|e(y_{k},z_{k})-e(y,z)|\le e(y_{k},y)+e(z_{k},z)$; the right side tends to $0$ by (a) and claim 1 of Arithmetic of Limits of Real Sequences, so the assertion follows from claim 3 of Order Properties of Limits of Real Sequences.

(c) Vectors. Let $H_{1},H_{2}$ be complex Hilbert spaces, let $v^{k}\to v$ and $w^{k}\to w$ in $H_{1}$ (for the metric of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces), and let $a,b$ be real. (i) $av^{k}+bw^{k}\to av+bw$: by claim 2 of The Induced Norm is a Norm, and Induces a Metric, $\lVert(av^{k}+bw^{k})-(av+bw)\rVert\le|a|\,\lVert v^{k}-v\rVert+|b|\,\lVert w^{k}-w\rVert$, whose right side tends to $0$ by (a) and claims 1 and 3 of Arithmetic of Limits of Real Sequences; conclude by claim 3 of Order Properties of Limits of Real Sequences and (a). By induction, finite real linear combinations of convergent sequences converge to the corresponding combination of the limits. (ii) $\lVert v^{k}\rVert\to\lVert v\rVert$, because $|\lVert v^{k}\rVert-\lVert v\rVert|\le\lVert v^{k}-v\rVert$ by the triangle inequality of claim 2 of The Induced Norm is a Norm, and Induces a Metric, and claim 3 of Order Properties of Limits of Real Sequences applies. (iii) If $\langle v^{k},w^{k}\rangle$ and $\langle v,w\rangle$ are real, then $\langle v^{k},w^{k}\rangle\to\langle v,w\rangle$. Indeed, sesquilinearity (Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces) gives

so by claim 7 of Properties of Complex Conjugation and Modulus and claim 1 of The Induced Norm is a Norm, and Induces a Metric its modulus is at most $\lVert v^{k}-v\rVert\,\lVert w^{k}-w\rVert+\lVert v^{k}-v\rVert\,\lVert w\rVert+\lVert v\rVert\,\lVert w^{k}-w\rVert$, which tends to $0$ by claims 1 and 2 of Arithmetic of Limits of Real Sequences; the modulus of a real number is its absolute value by claim 8 of Properties of Complex Conjugation and Modulus, and claim 3 of Order Properties of Limits of Real Sequences applies. (iv) If $V\in\mathcal{L}(H_{1},H_{2})$, then $Vv^{k}\to Vv$: by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §least-bound, $\lVert V\rVert_{\mathrm{op}}$ is a bound for $V$, so Bounded Linear Maps between Complex Normed Spaces and the Operator Norm §bounded gives $\lVert Vv^{k}-Vv\rVert=\lVert V(v^{k}-v)\rVert\le\lVert V\rVert_{\mathrm{op}}\lVert v^{k}-v\rVert$, and claim 3 of Arithmetic of Limits of Real Sequences and claim 3 of Order Properties of Limits of Real Sequences apply.

(d) Self-adjoint vectors. $\Omega\in H_{\mathrm{sa}}$, because $J\Omega=\Omega$ by The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §conjugation and Conjugation of a Complex Hilbert Space §fixed; and $\langle\xi,\eta\rangle$ is real for all $\xi,\eta\in H_{\mathrm{sa}}$ by Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §self-adjoint, where $J$ is the conjugation of $(H,M,\Omega)$ and $H_{\mathrm{sa}}=H^{J}$ as in Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws.

Step 1 (Approximating sequences). By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §limit there are sequences $(s^{k})_{k\in\mathbb{N}}$ and $(t^{k})_{k\in\mathbb{N}}$ of self-adjoint $d$-tuples in $M$ with $s^{k}_{j}\Omega\to X_{j}$ and $t^{k}_{j}\Omega\to Y_{j}$ for every $j\in[d]$; fix them, and put $\mu_{k}=\kappa_{d}(\lambda_{s^{k}})$ and $\nu_{k}=\kappa_{d}(\lambda_{t^{k}})$. By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §law, $\mu_{k}\to\mathrm{law}(X)$ and $\nu_{k}\to\mathrm{law}(Y)$ in $(\Sigma^{2}_{d},\widehat{W}_{2})$, which is a metric space 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.

Clause 1. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law, $s_{j}\Omega\in H_{\mathrm{sa}}$ for every $j$, so $s\Omega$ is an $L^{2}$ $d$-tuple by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples. The constant sequence with every term $s$ satisfies $\lVert s_{j}\Omega-s_{j}\Omega\rVert=0$, so it is an admissible sequence for $s\Omega$ in Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §law, and its sequence of laws is the constant sequence $\kappa_{d}(\lambda_{s})$, which converges to $\kappa_{d}(\lambda_{s})$ by condition 2 of Metric Space. Hence $\mathrm{law}(s\Omega)=\kappa_{d}(\lambda_{s})$.

Clause 2. For each $k$, Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling together with 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 (for the completion of $(\Sigma_{d},W_{2})$ in Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §laws) gives

By Step 0(b) and claim 2 of Arithmetic of Limits of Real Sequences, the left side tends to $\widehat{W}_{2}(\mathrm{law}(X),\mathrm{law}(Y))^{2}$. By Step 0(c)(i),(ii), $\lVert s^{k}_{j}\Omega-t^{k}_{j}\Omega\rVert\to\lVert X_{j}-Y_{j}\rVert$ for every $j$, so by claims 1 and 2 of Arithmetic of Limits of Real Sequences the right side tends to $\sum_{j=1}^{d}\lVert X_{j}-Y_{j}\rVert^{2}$, which is $\lVert X-Y\rVert_{2}^{2}$ by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples. Claim 1 of Order Properties of Limits of Real Sequences gives $\widehat{W}_{2}(\mathrm{law}(X),\mathrm{law}(Y))^{2}\le\lVert X-Y\rVert_{2}^{2}$. The two numbers squared here are nonnegative (condition 1 of Metric Space, and $\lVert X-Y\rVert_{2}$ is a nonnegative square root), so claim 2 of Monotonicity of Squaring on the Nonnegative Elements of an Ordered Field gives the assertion.

Clause 3. Reality. $\Omega$, $X_{i}$ and $X_{j}$ lie in $H_{\mathrm{sa}}$ (Step 0(d) and Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples), so $\langle\Omega,X_{i}\rangle$ and $\langle X_{i},X_{j}\rangle$ are real by Step 0(d).

First moments. By Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded (with $m=d$) and Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §moments, $\mathrm{m}_{i}(\mu_{k})=\lambda_{s^{k}}(x_{i})=\langle\Omega,s^{k}_{i}\Omega\rangle$, a real number; by Step 0(c)(iii), applied to the constant sequence $\Omega$ and to $(s^{k}_{i}\Omega)_{k}$, this sequence converges to $\langle\Omega,X_{i}\rangle$. On the other hand Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §lipschitz gives $|\mathrm{m}_{i}(\mu_{k})-\mathrm{m}_{i}(\mathrm{law}(X))|\le\widehat{W}_{2}(\mu_{k},\mathrm{law}(X))$, which tends to $0$ by Step 0(a), so $\mathrm{m}_{i}(\mu_{k})\to\mathrm{m}_{i}(\mathrm{law}(X))$ by claim 3 of Order Properties of Limits of Real Sequences. Uniqueness of real limits (Step 0(a)) gives $\mathrm{m}_{i}(\mathrm{law}(X))=\langle\Omega,X_{i}\rangle$.

Quadratic moments. Likewise $\mathrm{m}_{ij}(\mu_{k})=\lambda_{s^{k}}(x_{i}x_{j})=\langle s^{k}_{i}\Omega,s^{k}_{j}\Omega\rangle$, real by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §moments, and this sequence converges to $\langle X_{i},X_{j}\rangle$ by Step 0(c)(iii). Put $\omega_{k}=\widehat{W}_{2}(\mu_{k},\mathrm{law}(X))\ge0$, so $\omega_{k}\to0$ by Step 0(a), and let $\sigma$ be the nonnegative square root of $\widehat{M}(\mathrm{law}(X))\ge0$ (Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §moments). The third estimate of Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §lipschitz gives $\widehat{M}(\mu_{k})^{1/2}\le\sigma+\omega_{k}$, and its fourth estimate then gives

The right side tends to $0$ by claims 1 to 3 of Arithmetic of Limits of Real Sequences, so $\mathrm{m}_{ij}(\mu_{k})\to\mathrm{m}_{ij}(\mathrm{law}(X))$ by claim 3 of Order Properties of Limits of Real Sequences, and uniqueness of real limits gives $\mathrm{m}_{ij}(\mathrm{law}(X))=\langle X_{i},X_{j}\rangle$.

Second moment. By Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §moments, the formula just proved, Norm Induced by a Complex Inner Product and Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples,

Clause 4. Let $T=(A,c)$ be an affine datum from $d$ to $n$ variables. For $k\in\mathbb{N}$ let $u^{k}$ be the $n$-tuple with $u^{k}_{i}=c_{i}I+\sum_{j=1}^{d}A_{ij}s^{k}_{j}$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §affine, $u^{k}$ is a self-adjoint $n$-tuple in $M$, $u^{k}_{i}\Omega=c_{i}\Omega+\sum_{j=1}^{d}A_{ij}s^{k}_{j}\Omega$, and $\lambda_{u^{k}}=\lambda_{s^{k}}\circ\sigma_{T}$. By Step 0(c)(i), applied to the constant sequence $\Omega$ and the sequences $(s^{k}_{j}\Omega)_{k}$ with the real coefficients $c_{i},A_{ij}$, $u^{k}_{i}\Omega\to c_{i}\Omega+\sum_{j=1}^{d}A_{ij}X_{j}$, which is $(TX)_{i}$ by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations. Hence, by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §law for the $L^{2}$ $n$-tuple $TX$, $\kappa_{n}(\lambda_{u^{k}})\to\mathrm{law}(TX)$. By Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §bounded (with $m=d$), $\kappa_{n}(\lambda_{u^{k}})=\kappa_{n}(\lambda_{s^{k}}\circ\sigma_{T})=T_{\#}\mu_{k}$. By Calculus of Square-Integrable Noncommutative Laws: Agreement on Bounded Laws, Lipschitz Estimates, Functoriality of Push-Forwards, Moment Formulas, Positivity, the Cost and the Diagonal Coupling §lipschitz, $\widehat{W}_{2}(T_{\#}\mu_{k},T_{\#}\mathrm{law}(X))\le\lVert T\rVert\,\widehat{W}_{2}(\mu_{k},\mathrm{law}(X))$, whose right side tends to $0$ by Step 0(a) and claim 3 of Arithmetic of Limits of Real Sequences; so $T_{\#}\mu_{k}\to T_{\#}\mathrm{law}(X)$ by claim 3 of Order Properties of Limits of Real Sequences and Step 0(a). The sequence $(T_{\#}\mu_{k})_{k}$ thus converges in $(\Sigma^{2}_{n},\widehat{W}_{2})$ to both $\mathrm{law}(TX)$ and $T_{\#}\mathrm{law}(X)$, and these coincide by Uniqueness of Limits in a Metric Space.

Clause 5. The lemma is quantified over $d$, $n$ and the $L^{2}$ tuples, and the proof of Clause 4 above used only that $X$ is an $L^{2}$ tuple; so Clause 4 applies to the $L^{2}$ $2d$-tuple $(X,Y)$ of Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations and to affine data from $2d$ to $d$ variables. Likewise Clause 3 applies to every $L^{2}$ $d$-tuple.

Marginals. 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, $(\mathrm{pr}^{1}(X,Y))_{i}=0\,\Omega+\sum_{j=1}^{2d}P^{1}_{ij}(X,Y)_{j}=(X,Y)_{i}=X_{i}$ for $i\in[d]$, since $0\,\Omega$ is the zero vector; similarly $\mathrm{pr}^{2}=(P^{2},0)$ with $P^{2}_{ij}=1$ exactly when $j=d+i$, so $(\mathrm{pr}^{2}(X,Y))_{i}=(X,Y)_{d+i}=Y_{i}$. Hence $\mathrm{pr}^{1}(X,Y)=X$ and $\mathrm{pr}^{2}(X,Y)=Y$, and Clause 4 gives $\mathrm{pr}^{1}_{\#}\mathrm{law}(X,Y)=\mathrm{law}(X)$ and $\mathrm{pr}^{2}_{\#}\mathrm{law}(X,Y)=\mathrm{law}(Y)$. Since $\mathrm{law}(X,Y)\in\Sigma^{2}_{2d}$, Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §couplings gives $\mathrm{law}(X,Y)\in\Pi^{2}(\mathrm{law}(X),\mathrm{law}(Y))$.

Cost. The difference datum is $D=(P^{1}-P^{2},0)$ by Affine Data and Affine Substitutions of Noncommutative Polynomials §coordinate, so for $i\in[d]$

that is, $D(X,Y)=X-Y$ in the sense of Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples. By Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §cost, Clause 4 for $D$, and Clause 3 for the $L^{2}$ $d$-tuple $X-Y$,

Clause 6. Let $\pi$ be a trace-preserving embedding of $(H,M,\Omega)$ into $(K,N,\Psi)$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §embedding, for every $k$ the $d$-tuple $\pi(s^{k})$ is a self-adjoint $d$-tuple in $N$ with $\pi(s^{k}_{j})\Psi=V_{\pi}s^{k}_{j}\Omega$ and $\lambda_{\pi(s^{k})}=\lambda_{s^{k}}$. Since $V_{\pi}\in\mathcal{L}(H,K)$ by Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §isometry, Step 0(c)(iv) gives $\pi(s^{k}_{j})\Psi=V_{\pi}s^{k}_{j}\Omega\to V_{\pi}X_{j}$ for every $j\in[d]$. The $d$-tuple $V_{\pi}X$ is an $L^{2}$ $d$-tuple of $(K,N,\Psi)$ by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, so Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §law, applied in $(K,N,\Psi)$ with the sequence $(\pi(s^{k}))_{k}$, shows that $\mathrm{law}(V_{\pi}X)$ is the limit of $\kappa_{d}(\lambda_{\pi(s^{k})})=\kappa_{d}(\lambda_{s^{k}})=\mu_{k}$. This sequence converges to $\mathrm{law}(X)$ by Step 1, so $\mathrm{law}(V_{\pi}X)=\mathrm{law}(X)$ by Uniqueness of Limits in a Metric Space. $\blacksquare$