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

Each result cited is universally quantified over the data in its own statement. Laws $\lambda_{t}$ of self-adjoint tuples and resolvent transforms $\mathbf{R}(\cdot)$ are those of Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law and The Resolvent of a Self-Adjoint Vector and the Resolvent Transform of a Square-Integrable Tuple §transform; evaluation of polynomials obeys Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators. Fix once and for all a sequence $(P_{n'})_{n'\in\mathbb{N}}$ as in The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §recovery. For $r\in\mathbb{N}$ and an $l$-tuple $b$ in $\mathcal{P}_{r,\mathrm{sa}}$, $\sigma_{b}$ is its substitution.

Step 1 (realisation). 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 tracial W*-probability spaces $(H_{1},M_{1},\Omega_{1})$ and $(H_{2},M_{2},\Omega_{2})$, an $L^{2}$ $(k+m)$-tuple $W^{1}$ of the first with $\mathrm{law}(W^{1})=\pi_{1}$, and an $L^{2}$ $(k+n)$-tuple $W^{2}$ of the second with $\mathrm{law}(W^{2})=\pi_{2}$. Write $W^{1}=(X^{1},Y^{1})$ and $W^{2}=(X^{2},Z^{2})$ with $X^{1},X^{2}$ the $L^{2}$ $k$-tuples of the first $k$ entries (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations). By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations, $F^{m}W^{1}=X^{1}$ and $F^{n}W^{2}=X^{2}$ (the constant terms vanish), so by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward

Step 2 (bounded transforms with a common marginal). By The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the L^2 Lipschitz Bound and Universal Polynomial Recovery §consistent, $s^{1}=\mathbf{R}(W^{1})=(\mathbf{R}(X^{1}),\mathbf{R}(Y^{1}))$ is a self-adjoint $(2k+2m)$-tuple in $M_{1}$ and $s^{2}=\mathbf{R}(W^{2})=(\mathbf{R}(X^{2}),\mathbf{R}(Z^{2}))$ a self-adjoint $(2k+2n)$-tuple in $M_{2}$, the tuples in parentheses being concatenated. Put $\gamma_{1}=\lambda_{s^{1}}\in\Sigma_{2k+2m}$ and $\gamma_{2}=\lambda_{s^{2}}\in\Sigma_{2k+2n}$ (Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law), and let $a=(x_{1},\dots,x_{2k})$, regarded as a $2k$-tuple in $\mathcal{P}_{2k+2m,\mathrm{sa}}$ and in $\mathcal{P}_{2k+2n,\mathrm{sa}}$. For $p\in\mathcal{P}_{2k}$, $(\sigma_{a}p)(s^{1})=p(\mathbf{R}(X^{1}))$, so $\gamma_{1}\circ\sigma_{a}=\lambda_{\mathbf{R}(X^{1})}$; likewise $\gamma_{2}\circ\sigma_{a}=\lambda_{\mathbf{R}(X^{2})}$. By Step 1 and 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, $\lambda_{\mathbf{R}(X^{1})}=\lambda_{\mathbf{R}(X^{2})}=:\mu$.

Step 3 (amalgamated free product). The data $\gamma_{1},\gamma_{2}$, $a^{1}=a^{2}=a$, with the numbers $m_{1}=2k+2m$, $m_{2}=2k+2n$ and $2k$ in the roles of $m_{1},m_{2},n$ of Two Noncommutative Laws with a Common Marginal: Standing Notation for Their Amalgamated Free Product §data, satisfy that setting, with common marginal $\mu$. Let $\mathcal{H}$, $\Omega$, $\varphi$ and $\Lambda_{1},\Lambda_{2}$ be as in The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State, and let $L_{p}$ denote left multiplication on the complex GNS space of $\gamma_{1}$, of $\gamma_{2}$ or of $\mu$ according to the polynomial ring of $p$. Let $T$ be the $(2k+2m+2n)$-tuple in $\mathcal{L}(\mathcal{H})$

Each $L_{x_{i}}$ lies in the tracial algebra (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 (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), so by The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §representation every entry of $T$ is self-adjoint with norm at most $R$, where $R>0$ is a real number bounding the finitely many norms $\lVert L_{x_{i}}\rVert_{\mathrm{op}}$. Every $T_{w}$ ($w$ a word) is a finite product of operators $\Lambda_{f}(c)$ with $c\in A_{f}$; hence for words $u,v$ (the case of an empty word being trivial), rotating the factors of $T_{u}T_{v}$ one at a time with The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §trace gives $\langle\Omega,T_{u}T_{v}\Omega\rangle=\langle\Omega,T_{v}T_{u}\Omega\rangle$, where $\varphi(S)=\langle\Omega,S\Omega\rangle$ by The Amalgamated Free Product Space of Two Noncommutative Laws over a Common Marginal, Its Vacuum Vector and Vacuum State §state. By The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §marginals, $\lVert\Omega\rVert=1$, so The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State §law gives $\psi=\lambda_{T}\in\Sigma_{2k+2m+2n,R}$, where $\lambda_{T}(p)=\langle\Omega,p(T)\Omega\rangle$.

Let $b^{1}=(x_{1},\dots,x_{2k+2m})$ and $b^{2}=(x_{1},\dots,x_{2k},x_{2k+2m+1},\dots,x_{2k+2m+2n})$, tuples in $\mathcal{P}_{2k+2m+2n,\mathrm{sa}}$. For $p\in\mathcal{P}_{2k+2m}$, the transport rule Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §transport with $\Phi=\Lambda_{1}$ on the tracial algebra of $\gamma_{1}$, Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §gns, The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §marginals and 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 §trace give

For $i\le 2k$, Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §homomorphism gives $\pi_{\varepsilon}(L_{x_{i}})=L_{\sigma_{a}(x_{i})}=L_{x_{i}}$ (the left side taken on the GNS space of $\mu$), so $\Lambda_{1}(L_{x_{i}})=\Lambda_{2}(L_{x_{i}})$ by The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §amalgamation. Hence the entries of $T$ selected by $b^{2}$ are $\Lambda_{2}(L_{x_{1}}),\dots,\Lambda_{2}(L_{x_{2k+2n}})$, and the same computation with $\Lambda_{2}$ gives $\psi\circ\sigma_{b^{2}}=\gamma_{2}$.

Step 4 (a tracial W-probability space carrying $\psi$).* By The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star, $(H,M,\Omega_{\psi})=(\mathcal{H}_{\psi},\mathcal{M}_{\psi},\Omega_{\psi})$ is a tracial W*-probability space, and $u=(L_{x_{1}},\dots,L_{x_{2k+2m+2n}})$ is a self-adjoint tuple in $M$ with $\lambda_{u}=\psi$ (The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §law). Let $u^{A}$, $u^{B}$, $u^{C}$ be its blocks of entries $1,\dots,2k$, then $2k+1,\dots,2k+2m$, then $2k+2m+1,\dots,2k+2m+2n$. Since $p(u^{A},u^{B})=(\sigma_{b^{1}}p)(u)$ and $p(u^{A},u^{C})=(\sigma_{b^{2}}p)(u)$, Step 3 gives

Step 5 (reconstruction). Apply 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, with the fixed sequence $(P_{n'})$, to $W^{1}$ and the self-adjoint $2(k+m)$-tuple $(u^{A},u^{B})$: there is an $L^{2}$ $(k+m)$-tuple $\widehat{W}^{1}$ of $(H,M,\Omega_{\psi})$ with $\mathrm{law}(\widehat{W}^{1})=\mathrm{law}(W^{1})=\pi_{1}$, whose $j$-th entry is the limit of $P_{n'}(u_{2j-1},u_{2j})\Omega_{\psi}$ for $j\le k$. Apply it likewise to $W^{2}$ and $(u^{A},u^{C})$: there is an $L^{2}$ $(k+n)$-tuple $\widehat{W}^{2}$ with $\mathrm{law}(\widehat{W}^{2})=\pi_{2}$, whose $j$-th entry for $j\le k$ is the limit of the same sequence $P_{n'}(u_{2j-1},u_{2j})\Omega_{\psi}$. By Uniqueness of Limits in a Metric Space the first $k$ entries of $\widehat{W}^{1}$ and $\widehat{W}^{2}$ coincide; call this $L^{2}$ $k$-tuple $X$, and let $Y$ and $Z$ be the remaining entries of $\widehat{W}^{1}$ and $\widehat{W}^{2}$. Then $(X,Y)=\widehat{W}^{1}$ and $(X,Z)=\widehat{W}^{2}$, so $\mathrm{law}(X,Y)=\pi_{1}$ and $\mathrm{law}(X,Z)=\pi_{2}$.