Proof of Gluing Countably Many Square-Integrable Noncommutative Laws along a Common Marginal

Each result cited is universally quantified over the data in its own statement. Fix $k$, $\pi$, $(m_{j})_{j\in\mathbb{N}}$ and $(\gamma_{j})_{j\in\mathbb{N}}$ as in the statement. Laws $\lambda_{t}$ of self-adjoint tuples $t$ are those of Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law, resolvent transforms $\mathbf{R}(X)$ of $L^{2}$ tuples are those of The Resolvent of a Self-Adjoint Vector and the Resolvent Transform of a Square-Integrable Tuple §transform, and $p(t)$ is the value of a polynomial at a tuple of operators, with the rules of Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators. For a law $\lambda$, $\mathcal{H}_{\lambda}$, $\Omega_{\lambda}$, $\mathcal{M}_{\lambda}$, $\tau_{\lambda}$ and $L^{\lambda}_{p}$ are its complex GNS space, vacuum vector, tracial algebra, trace and left multiplication operators, as in Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §gns.

Notation. Put $d_{0}=k$ and, for $n\in\mathbb{N}$, $l_{n}=m_{1}+\dots+m_{n}$ and $d_{n}=k+l_{n}$; thus $d_{n+1}=d_{n}+m_{n+1}$ and $d_{n}\ge k+n>n$. For $n\in\mathbb{N}$ let $A^{n}=(G^{n},0)$ be the affine datum from $d_{n+1}$ to $d_{n}$ variables with $G^{n}_{ij}=1$ if $j=i$ and $0$ otherwise, and let $B^{n}=(Q^{n},0)$ be the affine datum from $d_{n}$ to $k+m_{n}$ variables with $Q^{n}_{ij}=1$ if either $i\le k$ and $j=i$, or $i>k$ and $j=d_{n-1}+i-k$, and $Q^{n}_{ij}=0$ otherwise. By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations (the constant terms vanish and each row has exactly one entry $1$), for every $L^{2}$ $d_{n+1}$-tuple $W$ the tuple $A^{n}W$ consists of the first $d_{n}$ entries of $W$; for every $L^{2}$ $d_{n}$-tuple $W$ the tuple $B^{n}W$ consists of the first $k$ entries of $W$ followed by its entries $d_{n-1}+1,\dots,d_{n}$; and for every $L^{2}$ $(k+l)$-tuple $W$ the tuple $F^{l}W$ consists of its first $k$ entries. We use Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward in the form $\mathrm{law}(TW)=T_{\#}\mathrm{law}(W)$ throughout.

Step 1 (consistent laws by dependent choice). Let $S$ be the set of pairs $(n,\Lambda)$ with $n\in\mathbb{N}$, $\Lambda\in\Sigma^{2}_{d_{n}}$ and $F^{l_{n}}_{\#}\Lambda=\pi$, a subset of $\mathbb{N}\times\bigcup_{d\in\mathbb{N}}\Sigma^{2}_{d}$. Let $\mathcal{R}$ be the relation on $S$ consisting of the pairs $((n,\Lambda),(n',\Lambda'))$ with $n'=n+1$, $A^{n}_{\#}\Lambda'=\Lambda$ and $B^{n+1}_{\#}\Lambda'=\gamma_{n+1}$. Since $l_{1}=m_{1}$ and $F^{m_{1}}_{\#}\gamma_{1}=\pi$, the pair $(1,\gamma_{1})$ lies in $S$.

Every $(n,\Lambda)\in S$ has an $\mathcal{R}$-successor in $S$. Indeed, $F^{l_{n}}_{\#}\Lambda=\pi=F^{m_{n+1}}_{\#}\gamma_{n+1}$, so Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal §glue, applied with $k$, $l_{n}$, $m_{n+1}$, $\Lambda$ and $\gamma_{n+1}$ in place of its $k$, $m$, $n$, $\pi_{1}$ and $\pi_{2}$, gives a tracial W*-probability space with an $L^{2}$ $k$-tuple $X$, an $L^{2}$ $l_{n}$-tuple $Y$ and an $L^{2}$ $m_{n+1}$-tuple $Z'$ such that $\mathrm{law}(X,Y)=\Lambda$ and $\mathrm{law}(X,Z')=\gamma_{n+1}$. Put $W=(X,Y,Z')$, an $L^{2}$ $d_{n+1}$-tuple, and $\Lambda'=\mathrm{law}(W)\in\Sigma^{2}_{d_{n+1}}$. By the Notation, $A^{n}W=(X,Y)$, $B^{n+1}W=(X,Z')$, $F^{l_{n+1}}W=X$ and $F^{l_{n}}(X,Y)=X$; hence $A^{n}_{\#}\Lambda'=\mathrm{law}(X,Y)=\Lambda$, $B^{n+1}_{\#}\Lambda'=\mathrm{law}(X,Z')=\gamma_{n+1}$, and $F^{l_{n+1}}_{\#}\Lambda'=\mathrm{law}(X)=F^{l_{n}}_{\#}\mathrm{law}(X,Y)=F^{l_{n}}_{\#}\Lambda=\pi$. So $(n+1,\Lambda')\in S$ and it is an $\mathcal{R}$-successor of $(n,\Lambda)$.

By Axiom of Dependent Choice there is a sequence $(a_{n})_{n\in\mathbb{N}}$ in $S$ with $a_{1}=(1,\gamma_{1})$ and $(a_{n},a_{n+1})\in\mathcal{R}$ for every $n$. By induction the first component of $a_{n}$ is $n$; write $a_{n}=(n,\Lambda_{n})$. Then for every $n\in\mathbb{N}$:

$$\Lambda_{1}=\gamma_{1},\qquad\Lambda_{n}\in\Sigma^{2}_{d_{n}},\qquad A^{n}_{\#}\Lambda_{n+1}=\Lambda_{n},\qquad B^{n+1}_{\#}\Lambda_{n+1}=\gamma_{n+1}.\qquad(1.1)$$

Step 2 (the bounded laws). Fix $n\in\mathbb{N}$. By Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §law there are a tracial W*-probability space $(K,N,\Psi)$ and an $L^{2}$ $d_{n}$-tuple $X^{0}$ of it with $\mathrm{law}(X^{0})=\Lambda_{n}$. By The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §consistent, $\mathbf{R}(X^{0})$ is a self-adjoint $2d_{n}$-tuple in $N$, so $\Gamma_{n}=\lambda_{\mathbf{R}(X^{0})}\in\Sigma_{2d_{n}}$ by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law. By The Law of the Resolvent Transform is a Lipschitz Function of the Law, and Reconstruction of a Square-Integrable Tuple from a Bounded Tuple with the Law of Its Transform §law, $\lambda_{\mathbf{R}(X)}=\Gamma_{n}$ for every $L^{2}$ $d_{n}$-tuple $X$ of every tracial W*-probability space with $\mathrm{law}(X)=\Lambda_{n}$; in particular $\Gamma_{n}$ does not depend on the choice of $X^{0}$, and is determined by $\Lambda_{n}$.

For $n\in\mathbb{N}$ let $\beta_{n}=\sigma_{(x_{1},\dots,x_{2d_{n}})}:\mathcal{P}_{2d_{n}}\to\mathcal{P}_{2d_{n+1}}$ be the substitution of the first $2d_{n}$ variables. We claim

$$\Gamma_{n+1}\circ\beta_{n}=\Gamma_{n}.\qquad(2.1)$$

Take, by Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §law, an $L^{2}$ $d_{n+1}$-tuple $X^{0}$ of a tracial W*-probability space $(K,N,\Psi)$ with $\mathrm{law}(X^{0})=\Lambda_{n+1}$, and write $X^{0}=(X',X'')$ with $X'=A^{n}X^{0}$ its first $d_{n}$ entries. By The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §consistent, the first $2d_{n}$ entries of $\mathbf{R}(X^{0})$ form $\mathbf{R}(X')$. For $p\in\mathcal{P}_{2d_{n}}$, Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values give $(\beta_{n}p)(\mathbf{R}(X^{0}))=p(\mathbf{R}(X'))$, so by Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law, $\Gamma_{n+1}(\beta_{n}p)=\langle\Psi,p(\mathbf{R}(X'))\Psi\rangle=\lambda_{\mathbf{R}(X')}(p)$. Finally $\mathrm{law}(X')=A^{n}_{\#}\Lambda_{n+1}=\Lambda_{n}$ by (1.1), so $\lambda_{\mathbf{R}(X')}=\Gamma_{n}$ by the previous paragraph.

Step 3 (a chain of tracial W-probability spaces).* For $n\in\mathbb{N}$ write $H_{n}=\mathcal{H}_{\Gamma_{n}}$, $M_{n}=\mathcal{M}_{\Gamma_{n}}$, $\Omega_{n}=\Omega_{\Gamma_{n}}$ and $L^{(n)}_{p}=L^{\Gamma_{n}}_{p}$ for $p\in\mathcal{P}_{2d_{n}}$. By The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star, $(H_{n},M_{n},\Omega_{n})$ is a tracial W*-probability space with trace $\tau_{\Gamma_{n}}$. Every variable $x_{i}$ is self-adjoint by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint and Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, so $L^{(n)}_{x_{i}}$ 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, and it lies in $M_{n}$ 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.

Let $\theta_{n}:M_{n}\to M_{n+1}$ be the map $\pi$ of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §embedding, with $2d_{n+1}$, $2d_{n}$, $\Gamma_{n+1}$ and the $2d_{n}$-tuple $(x_{1},\dots,x_{2d_{n}})$ in $\mathcal{P}_{2d_{n+1},\mathrm{sa}}$ in place of its $m$, $n$, $\gamma$ and $a$; its substitution $\sigma_{a}$ is $\beta_{n}$, so its marginal law $\gamma\circ\sigma_{a}$ is $\Gamma_{n}$ by (2.1), and $\theta_{n}$ maps $M_{n}$ into $M_{n+1}$. By Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §homomorphism, $\theta_{n}$ is linear, unital, multiplicative, $*$-preserving and satisfies $\tau_{\Gamma_{n+1}}\circ\theta_{n}=\tau_{\Gamma_{n}}$, so it is a trace-preserving embedding of $(H_{n},M_{n},\Omega_{n})$ into $(H_{n+1},M_{n+1},\Omega_{n+1})$ in the sense of Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §embedding; and the same clause with Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values gives

$$\theta_{n}(L^{(n)}_{x_{i}})=L^{(n+1)}_{\beta_{n}(x_{i})}=L^{(n+1)}_{x_{i}}\qquad(i\in[2d_{n}]).\qquad(3.1)$$

Step 4 (the inductive limit and one sequence of operators). Apply Inductive Limits of Tracial W*-Probability Spaces along Trace-Preserving Embeddings with $\theta_{n}$ in place of its $\pi_{k}$: there are a tracial W*-probability space $(H,M,\Omega)$ and trace-preserving embeddings $\rho_{n}$ of $(H_{n},M_{n},\Omega_{n})$ into it with $\rho_{n+1}(\theta_{n}(T))=\rho_{n}(T)$ for all $n$ and $T\in M_{n}$, by Inductive Limits of Tracial W*-Probability Spaces along Trace-Preserving Embeddings §compatible. Since $(d_{n})$ is increasing, if $i\le2d_{n}$ then $i\le2d_{n'}$ for all $n'\ge n$, and induction on $n'$ using (3.1) and compatibility gives $\rho_{n'}(L^{(n')}_{x_{i}})=\rho_{n}(L^{(n)}_{x_{i}})$ for all $n'\ge n$. Hence, for $i\in\mathbb{N}$, the element

$$S_{i}=\rho_{n}(L^{(n)}_{x_{i}})\in M$$

is the same for every $n\in\mathbb{N}$ with $i\le2d_{n}$ (compare two such indices through the smaller one); such $n$ exist since $d_{i}>i$.

Fix $n\in\mathbb{N}$ and let $u^{n}=(L^{(n)}_{x_{1}},\dots,L^{(n)}_{x_{2d_{n}}})$, a self-adjoint $2d_{n}$-tuple in $M_{n}$ by Step 3. By The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §law and Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law, $\lambda_{u^{n}}=\Gamma_{n}$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §embedding, $s^{n}=(\rho_{n}(u^{n}_{1}),\dots,\rho_{n}(u^{n}_{2d_{n}}))=(S_{1},\dots,S_{2d_{n}})$ is a self-adjoint $2d_{n}$-tuple in $M$ with

$$\lambda_{s^{n}}=\lambda_{u^{n}}=\Gamma_{n}.\qquad(4.1)$$

Step 5 (reconstruction of all blocks at once). Fix, once and for all, a sequence $(P_{r})_{r\in\mathbb{N}}$ as in The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §recovery. Fix $n\in\mathbb{N}$, and take by Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §law an $L^{2}$ $d_{n}$-tuple $X^{0}$ of some tracial W*-probability space with $\mathrm{law}(X^{0})=\Lambda_{n}$; by Step 2 and (4.1), $\lambda_{s^{n}}=\Gamma_{n}=\lambda_{\mathbf{R}(X^{0})}$. So The Law of the Resolvent Transform is a Lipschitz Function of the Law, and Reconstruction of a Square-Integrable Tuple from a Bounded Tuple with the Law of Its Transform §reconstruction, applied with $d_{n}$, $X^{0}$, the space $(H,M,\Omega)$ in place of its $(K,N,\Psi)$, and $s^{n}$ in place of its $s$, shows: for every $j\in[d_{n}]$ the sequence $(P_{r}(S_{2j-1},S_{2j})\Omega)_{r\in\mathbb{N}}$ converges in $H$ to a vector fixed by the conjugation of $(H,M,\Omega)$, and the $d_{n}$-tuple of these limits is an $L^{2}$ $d_{n}$-tuple of $(H,M,\Omega)$ with law $\Lambda_{n}$.

The sequence $(P_{r}(S_{2j-1},S_{2j})\Omega)_{r}$ does not involve $n$. For $j\in\mathbb{N}$ let $V_{j}\in H$ be its limit, which exists by the preceding paragraph applied with $n=j$ (as $j\le d_{j}$) and is unique by Uniqueness of Limits in a Metric Space. Then, for every $n\in\mathbb{N}$, $\widehat{W}^{n}=(V_{1},\dots,V_{d_{n}})$ is an $L^{2}$ $d_{n}$-tuple of $(H,M,\Omega)$ with

$$\mathrm{law}(\widehat{W}^{n})=\Lambda_{n}.\qquad(5.1)$$

Step 6 (the tuples). Let $Z=(V_{1},\dots,V_{k})$, an $L^{2}$ $k$-tuple of $(H,M,\Omega)$, and for $j\in\mathbb{N}$ let $Y_{j}=(V_{d_{j-1}+1},\dots,V_{d_{j}})$, an $L^{2}$ $m_{j}$-tuple of $(H,M,\Omega)$, since $d_{j}-d_{j-1}=m_{j}$. For $j=1$ we have $d_{0}=k$, so $(Z,Y_{1})=\widehat{W}^{1}$ and $\mathrm{law}(Z,Y_{1})=\Lambda_{1}=\gamma_{1}$ by (5.1) and (1.1). For $j\ge2$, write $j=n+1$ with $n\in\mathbb{N}$; by the Notation $(Z,Y_{j})=B^{j}\widehat{W}^{j}$, so by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, (5.1) and (1.1),

$$\mathrm{law}(Z,Y_{j})=B^{j}_{\#}\mathrm{law}(\widehat{W}^{j})=B^{n+1}_{\#}\Lambda_{n+1}=\gamma_{j}.$$

Thus $(H,M,\Omega)$, $Z$ and $(Y_{j})_{j\in\mathbb{N}}$ have the required properties.