Proof of Gluing Two Noncommutative Couplings along a Common Marginal

Each result cited below is universally quantified over the data in its own statement. We adopt the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, together with the Hilbert space conventions of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, and fix $d$, $R$, $\mu,\nu,\rho$, $\gamma_{1},\gamma_{2}$ and $\sigma^{12},\sigma^{23},\sigma^{13}$ as in the statement.

Step 0 (the amalgamation data). By Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §bound, $\gamma_{1}\in\Pi(\mu,\nu)$ and $\gamma_{2}\in\Pi(\nu,\rho)$ lie in $\Sigma_{2d,R}$, hence in $\Sigma_{2d}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law. Put $m_{1}=m_{2}=2d$ and $n=d$, and let

$d$-tuples in $\mathcal{P}_{2d}$ whose entries are variables and hence lie in $\mathcal{P}_{2d,\mathrm{sa}}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint. They are exactly the tuples defining the marginal substitutions in Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals, so $\sigma_{a^{1}}=\iota^{2}$ and $\sigma_{a^{2}}=\iota^{1}$, and Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling gives

Thus the data of Two Noncommutative Laws with a Common Marginal: Standing Notation for Their Amalgamated Free Product §data hold for $\gamma_{1},\gamma_{2},a^{1},a^{2}$. The letters clash: the common marginal, which that setting (and every result stated in it) calls $\mu$, is here $\nu$; our $\mu$ plays no role in the amalgamation. With this renaming we use the notation of its algebras and its embeddings: $N=\mathcal{M}_{\nu}$ acting on $\mathcal{H}_{\nu}$, $A_{\varepsilon}=\mathcal{M}_{\gamma_{\varepsilon}}$ acting on $\mathcal{H}_{\gamma_{\varepsilon}}$ with trace $\tau_{\gamma_{\varepsilon}}$, and the embeddings $\pi_{\varepsilon}:N\to A_{\varepsilon}$. Let $\mathcal{H}$, $\Omega$, $\varphi$ and $\Lambda_{\varepsilon}$ be the amalgamated free product space, its vacuum vector, its vacuum state and the actions, as in The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State; by The Amalgamated Free Product Space of Two Noncommutative Laws over a Common Marginal, Its Vacuum Vector and Vacuum State §state, $\varphi(X)=\langle\Omega,X\Omega\rangle_{\mathcal{H}}$ for $X\in\mathcal{L}(\mathcal{H})$.

For $q\in\mathcal{P}_{2d}$ let $L^{\varepsilon}_{q}$ be the left multiplication operator of 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 on $\mathcal{H}_{\gamma_{\varepsilon}}$, and for $q\in\mathcal{P}_{d}$ let $L^{\nu}_{q}$ be the one on $\mathcal{H}_{\nu}$. 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, $L^{\varepsilon}_{q}\in A_{\varepsilon}$ and $L^{\nu}_{q}\in N$. Since $\gamma_{\varepsilon}\in\Sigma_{2d,R}$, 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, with $2d$ and $R$ in place of that lemma's $d$ and $r$, gives $\lVert L^{\varepsilon}_{x_{i}}\rVert_{\mathrm{op}}\le R$ for $i\in[2d]$, and 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 together with $x_{i}^{*}=x_{i}$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint) gives $(L^{\varepsilon}_{x_{i}})^{*}=L^{\varepsilon}_{x_{i}}$.

Step 1 (the operators). Define $f:[3d]\to\{1,2\}$ and $c_{i}\in A_{f(i)}$ by $f(i)=1$, $c_{i}=L^{1}_{x_{i}}$ for $i\in[2d]$, and $f(2d+j)=2$, $c_{2d+j}=L^{2}_{x_{d+j}}$ for $j\in[d]$, and let $S=(S_{1},\dots,S_{3d})$ be the $3d$-tuple in $\mathcal{L}(\mathcal{H})$ with $S_{i}=\Lambda_{f(i)}(c_{i})$. We claim that $S_{d+j}=\Lambda_{2}(L^{2}_{x_{j}})$ for $j\in[d]$. By Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §homomorphism, applied with $\gamma_{\varepsilon}$, $a^{\varepsilon}$, $2d$, $d$ in place of that lemma's $\gamma$, $a$, $m$, $n$ (its $\mu$ being our $\nu$), and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values,

Since $L^{\nu}_{x_{j}}\in N$, The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §amalgamation gives $S_{d+j}=\Lambda_{1}(\pi_{1}(L^{\nu}_{x_{j}}))=\Lambda_{2}(\pi_{2}(L^{\nu}_{x_{j}}))=\Lambda_{2}(L^{2}_{x_{j}})$. Consequently

Each $c_{i}$ is some $L^{\varepsilon}_{x_{k}}$ with $k\in[2d]$, so by The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §representation and Step 0, $S_{i}^{*}=\Lambda_{f(i)}(c_{i}^{*})=\Lambda_{f(i)}(c_{i})=S_{i}$ and $\lVert S_{i}\rVert_{\mathrm{op}}\le\lVert c_{i}\rVert_{\mathrm{op}}\le R$ for every $i\in[3d]$.

Step 2 (the glued state). By The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §marginals, $\lVert\Omega\rVert_{\mathcal{H}}=1$. By Step 1, The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State applies with $3d$, $\mathcal{H}$, $S$ in place of its $n$, $H$, $T$; let $\psi=\lambda_{S}$, so that $\psi(p)=\langle\Omega,p(S)\Omega\rangle=\varphi(p(S))$ for $p\in\mathcal{P}_{3d}$. To get $\psi\in\Sigma_{3d,R}$ from The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State §law we show $\varphi(S_{u}S_{v})=\varphi(S_{v}S_{u})$ for all $u,v\in W_{3d}$. If $u=\varnothing$ then $S_{u}=I$ by Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §word-products, and both sides equal $\varphi(S_{v})$; likewise if $v=\varnothing$. If $u,v$ are nonempty, then $S_{u}S_{v}=S_{uv}$ and $S_{v}S_{u}=S_{vu}$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, so it suffices to prove

First, for a letter $(i)$ and a nonempty word $v$ of length $l$ with letters $v_{1},\dots,v_{l}$,

Indeed, $S_{(i)}=S_{i}$ and $S_{v}=S_{v_{1}}\cdots S_{v_{l}}$ by Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §word-products, and $S_{(i)v}=S_{i}S_{v}$, $S_{v(i)}=S_{v}S_{i}$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values. As composition is associative, $S_{(i)v}$ is the product $\Lambda_{f(i)}(c_{i})\Lambda_{f(v_{1})}(c_{v_{1}})\cdots\Lambda_{f(v_{l})}(c_{v_{l}})$ of $l+1\ge2$ factors and $S_{v(i)}$ is the product of the same factors with the first one moved to the end, so (3) is The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §trace with $r=l+1$.

We prove (2) by induction on the length $k\in\mathbb{N}$ of $u$, for all nonempty $v$ simultaneously. If $k=1$, then $u$ is a letter by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter, and (2) is (3). Suppose (2) holds for all words $u$ of length $k$, and let $u$ have length $k+1$. By Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter, $u=u'(i)$ with $u'$ of length $k$, hence nonempty. By Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §monoid, $uv=u'\bigl((i)v\bigr)$, the word $(i)v$ is nonempty, $\bigl((i)v\bigr)u'=(i)(vu')$ with $vu'$ nonempty, and $(vu')(i)=v\bigl(u'(i)\bigr)=vu$. The induction hypothesis (for $u'$ and $(i)v$) and then (3) (for $i$ and $vu'$) give

This proves (2), and hence $\psi\in\Sigma_{3d,R}$ by The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State §law.

Step 3 (the first and second pairs). Let $p\in\mathcal{P}_{2d}$ and let $a=(x_{1},\dots,x_{2d})$, the tuple in $\mathcal{P}_{3d}$ defining $\sigma^{12}$. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, $a(S)=(x_{1}(S),\dots,x_{2d}(S))=(S_{1},\dots,S_{2d})$, so by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution and (1),

Apply Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §transport with $\mathcal{H}_{\gamma_{1}}$, $\mathcal{H}$, $2d$, $(L^{1}_{x_{1}},\dots,L^{1}_{x_{2d}})$, $A_{1}$, $\Lambda_{1}$ in place of its $H$, $K$, $n$, $T$, $\mathcal{A}$, $\Phi$: the set $A_{1}$ contains $I$ and every $L^{1}_{x_{i}}$ and is closed under sums, complex multiples and composition 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 $\Lambda_{1}$ is linear, multiplicative and unital by The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §representation. Together with Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §gns (for $\gamma_{1}\in\Sigma_{2d}$), which gives $p(L^{1}_{x_{1}},\dots,L^{1}_{x_{2d}})=L^{1}_{p}$, this yields $\bigl(\sigma^{12}(p)\bigr)(S)=\Lambda_{1}(L^{1}_{p})$. By 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,

Hence $\psi\circ\sigma^{12}=\gamma_{1}$, which is claim 1. The same argument with the tuple $(x_{d+1},\dots,x_{3d})$ defining $\sigma^{23}$, for which $a(S)=(S_{d+1},\dots,S_{3d})=\bigl(\Lambda_{2}(L^{2}_{x_{1}}),\dots,\Lambda_{2}(L^{2}_{x_{2d}})\bigr)$ by (1), and with $\mathcal{H}_{\gamma_{2}}$, $A_{2}$, $\Lambda_{2}$, $\gamma_{2}$ in place of $\mathcal{H}_{\gamma_{1}}$, $A_{1}$, $\Lambda_{1}$, $\gamma_{1}$, gives $\psi\circ\sigma^{23}=\gamma_{2}$, which is claim 2.

Step 4 (the composite coupling). Write $\gamma_{3}=\psi\circ\sigma^{13}:\mathcal{P}_{2d}\to\mathbb{C}$. It is linear, as $\psi$ is linear by The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State §state and $\sigma^{13}$ is linear by Substitution of Noncommutative Polynomials into the Variables §substitution. The entries of the tuple defining $\sigma^{13}$ are variables, hence self-adjoint by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint. For $p,q\in\mathcal{P}_{2d}$, Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values, Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §homomorphism and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §adjoint give $\sigma^{13}(1)=1$, $\sigma^{13}(pq)=\sigma^{13}(p)\sigma^{13}(q)$ and $\sigma^{13}(p^{*}p)=\sigma^{13}(p)^{*}\sigma^{13}(p)$. Since $\psi\in\Sigma_{3d,R}$ is a tracial state (Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state), it follows that $\gamma_{3}(1)=1$, that $\gamma_{3}(p^{*}p)$ is real and nonnegative, and that $\gamma_{3}(pq)=\psi\bigl(\sigma^{13}(p)\sigma^{13}(q)\bigr)=\psi\bigl(\sigma^{13}(q)\sigma^{13}(p)\bigr)=\gamma_{3}(qp)$; so $\gamma_{3}$ is a tracial state on $\mathcal{P}_{2d}$.

For the marginals, apply Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §composition with $\iota^{1}=\sigma_{(x_{1},\dots,x_{d})}$ as the inner substitution. For $j\in[d]$, the $j$-th entries of the tuples defining $\sigma^{13}$ and $\sigma^{12}$ are both $x_{j}$, so by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values $\sigma^{13}(x_{j})=\sigma^{12}(x_{j})=x_{j}$; hence $\sigma^{13}\circ\iota^{1}$ and $\sigma^{12}\circ\iota^{1}$ are both the substitution of $(x_{1},\dots,x_{d})$ in $\mathcal{P}_{3d}$, and by claim 1 and Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling (for $\gamma_{1}\in\Pi(\mu,\nu)$)

Likewise, with $\iota^{2}=\sigma_{(x_{d+1},\dots,x_{2d})}$ as the inner substitution: for $j\in[d]$ the $(d+j)$-th entries of the tuples defining $\sigma^{13}$ and $\sigma^{23}$ are both $x_{2d+j}$, so $\sigma^{13}\circ\iota^{2}$ and $\sigma^{23}\circ\iota^{2}$ are both the substitution of $(x_{2d+1},\dots,x_{3d})$, and by claim 2 and Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling (for $\gamma_{2}\in\Pi(\nu,\rho)$)

As $\mu,\rho\in\Sigma_{d}$, Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling gives $\gamma_{3}\in\Pi(\mu,\rho)$, which is claim 3.