Proof of Gluing Countably Many Noncommutative Couplings with a Common First Marginal in One Tracial W*-Probability Space

Each result cited is universally quantified over the data in its own statement. Fix $d$, $R$, $\mu$, $(\nu_{k})_{k\in\mathbb{N}}$ and $(\gamma_{k})_{k\in\mathbb{N}}$ as in the statement.

Notation. For a law $\lambda$ and a polynomial $p$ in its variables, $L^{\lambda}_{p}$ is the left multiplication operator on the complex GNS space $\mathcal{H}_{\lambda}$ given 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 §multiplication; $\Omega_{\lambda}$, $\mathcal{M}_{\lambda}$ and $\tau_{\lambda}$ are the vacuum vector, tracial algebra and trace, as in Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §gns. The letter $I$ denotes an identity operator. For $k\in\mathbb{N}$, with $kd$ the product of $k$ and $d$, let

$$\beta_{k}=\sigma_{(x_{1},\dots,x_{(k+1)d})}:\mathcal{P}_{(k+1)d}\to\mathcal{P}_{(k+2)d},\qquad\delta_{k}=\sigma_{(x_{1},\dots,x_{d},x_{kd+1},\dots,x_{(k+1)d})}:\mathcal{P}_{2d}\to\mathcal{P}_{(k+1)d},\qquad\alpha_{k}=\sigma_{(x_{1},\dots,x_{d})}:\mathcal{P}_{d}\to\mathcal{P}_{(k+1)d}$$

be the substitutions of the indicated tuples of variables. A map $\Gamma$ satisfies $(B_{k})$ if $\Gamma\in\Sigma_{(k+1)d,R}$ and $\Gamma\circ\delta_{k}=\gamma_{k}$. For $l\in\mathbb{N}$, every variable $x_{i}$ of $\mathcal{P}_{l}$ lies in $\mathcal{P}_{l,\mathrm{sa}}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint and satisfies $x_{i}^{*}=x_{i}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint; hence, for every $\lambda\in\Sigma_{l}$ (which lies in some $\Sigma_{l,r}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law) and $i\in[l]$, 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 gives $(L^{\lambda}_{x_{i}})^{*}=L^{\lambda}_{x_{i}}$ and that $L^{\lambda}_{x_{i}}$ is self-adjoint.

Step 1 (the couplings and the first block). Let $k\in\mathbb{N}$. Since $\mu,\nu_{k}\in\Sigma_{d,R}$, Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §bound gives $\gamma_{k}\in\Sigma_{2d,R}$, so $\gamma_{k}\in\Sigma_{2d}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law. By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §composition, $\delta_{k}\circ\iota^{1}=\sigma_{c}$ with $c_{j}=\delta_{k}(x_{j})$ for $j\in[d]$, and $\delta_{k}(x_{j})=x_{j}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values (the first $d$ entries of the tuple of $\delta_{k}$ are $x_{1},\dots,x_{d}$); thus $\delta_{k}\circ\iota^{1}=\alpha_{k}$, where $\iota^{1}=\sigma_{(x_{1},\dots,x_{d})}$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals. Consequently, if $\Gamma$ satisfies $(B_{k})$, then by Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling (for $\gamma_{k}\in\Pi(\mu,\nu_{k})$)

$$\Gamma\circ\alpha_{k}=\Gamma\circ\delta_{k}\circ\iota^{1}=\gamma_{k}\circ\iota^{1}=\mu.\qquad(1.1)$$

Step 2 (one gluing step: the amalgamation data). Fix $k\in\mathbb{N}$ and a map $\Gamma$ satisfying $(B_{k})$. Take the data of Two Noncommutative Laws with a Common Marginal: Standing Notation for Their Amalgamated Free Product §data with $n=d$; with $m_{1}=(k+1)d$, first law $\Gamma$ and the $d$-tuple $a^{1}=(x_{1},\dots,x_{d})$ in $\mathcal{P}_{(k+1)d,\mathrm{sa}}$; and with $m_{2}=2d$, second law $\gamma_{k+1}$ and the $d$-tuple $a^{2}=(x_{1},\dots,x_{d})$ in $\mathcal{P}_{2d,\mathrm{sa}}$. Here $\Gamma\in\Sigma_{(k+1)d}$ by $(B_{k})$ and Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law, and $\gamma_{k+1}\in\Sigma_{2d}$ by Step 1. The common marginal condition holds with marginal $\mu$: $\sigma_{a^{1}}=\alpha_{k}$, so $\Gamma\circ\sigma_{a^{1}}=\mu$ by (1.1); and $\sigma_{a^{2}}=\iota^{1}$, so $\gamma_{k+1}\circ\sigma_{a^{2}}=\mu$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling. The two laws which that setting calls $\gamma_{1},\gamma_{2}$ are here $\Gamma$ and $\gamma_{k+1}$ (our couplings $\gamma_{1},\gamma_{2}$ play no role in this step); its common marginal is our $\mu$. In the notation of Two Noncommutative Laws with a Common Marginal: Standing Notation for Their Amalgamated Free Product §algebras and Two Noncommutative Laws with a Common Marginal: Standing Notation for Their Amalgamated Free Product §embeddings: $N=\mathcal{M}_{\mu}$, $A_{1}=\mathcal{M}_{\Gamma}$, $A_{2}=\mathcal{M}_{\gamma_{k+1}}$, and $\pi_{1}:N\to A_{1}$, $\pi_{2}:N\to A_{2}$ are the embeddings of Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §embedding for $(\Gamma,a^{1})$ and $(\gamma_{k+1},a^{2})$. Let $\mathcal{H}$, $\Omega$ and $\varphi$ be the amalgamated free product space, its vacuum vector and its vacuum state (The Amalgamated Free Product Space of Two Noncommutative Laws over a Common Marginal, Its Vacuum Vector and Vacuum State §space, The Amalgamated Free Product Space of Two Noncommutative Laws over a Common Marginal, Its Vacuum Vector and Vacuum State §vectors, The Amalgamated Free Product Space of Two Noncommutative Laws over a Common Marginal, Its Vacuum Vector and Vacuum State §state), so that $\varphi(X)=\langle\Omega,X\Omega\rangle_{\mathcal{H}}$ for $X\in\mathcal{L}(\mathcal{H})$, and let $\Lambda_{1},\Lambda_{2}$ be the actions of The Amalgamated Free Product Space: Multilinearity of the Tuple Vectors and the Bounded Left Actions of the Two Tracial Algebras §actions; The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State applies to them. All these objects are determined by $(k,\Gamma)$ through the cited constructions; no choice is made.

For $j\in[d]$, $L^{\mu}_{x_{j}}\in N$ 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, and Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §homomorphism with Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values gives $\pi_{1}(L^{\mu}_{x_{j}})=L^{\Gamma}_{\sigma_{a^{1}}(x_{j})}=L^{\Gamma}_{x_{j}}$ and $\pi_{2}(L^{\mu}_{x_{j}})=L^{\gamma_{k+1}}_{\sigma_{a^{2}}(x_{j})}=L^{\gamma_{k+1}}_{x_{j}}$. Hence The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §amalgamation yields

$$\Lambda_{1}(L^{\Gamma}_{x_{j}})=\Lambda_{1}(\pi_{1}(L^{\mu}_{x_{j}}))=\Lambda_{2}(\pi_{2}(L^{\mu}_{x_{j}}))=\Lambda_{2}(L^{\gamma_{k+1}}_{x_{j}})\qquad(j\in[d]).\qquad(2.1)$$

Step 3 (one gluing step: the glued law). Keep the data of Step 2 and put $m=(k+2)d$. Define $f:[m]\to\{1,2\}$ and $c_{i}\in A_{f(i)}$ by $f(i)=1$, $c_{i}=L^{\Gamma}_{x_{i}}$ for $i\in[(k+1)d]$, and $f((k+1)d+j)=2$, $c_{(k+1)d+j}=L^{\gamma_{k+1}}_{x_{d+j}}$ for $j\in[d]$; these operators lie in $A_{1}$, respectively $A_{2}$, 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 $T$ be the $m$-tuple in $\mathcal{L}(\mathcal{H})$ with $T_{i}=\Lambda_{f(i)}(c_{i})$.

(a) Self-adjointness and norms. By Notation, $c_{i}^{*}=c_{i}$, so $T_{i}^{*}=\Lambda_{f(i)}(c_{i}^{*})=T_{i}$ by The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §representation; by Adjoint of a Linear Map between Complex Inner Product Spaces §adjoint this means $\langle T_{i}\xi,\eta\rangle=\langle\xi,T_{i}\eta\rangle$ for all $\xi,\eta\in\mathcal{H}$, that is, $T_{i}$ is self-adjoint in the sense of Self-Adjoint Operator. Since $\Gamma\in\Sigma_{(k+1)d,R}$ and $\gamma_{k+1}\in\Sigma_{2d,R}$, 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 $(k+1)d$, respectively $2d$, and $R$ in place of its $d$ and $r$) gives $\lVert c_{i}\rVert_{\mathrm{op}}\le R$, so $\lVert T_{i}\rVert_{\mathrm{op}}\le\lVert c_{i}\rVert_{\mathrm{op}}\le R$ by The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §representation, for every $i\in[m]$.

(b) Traciality. We show $\varphi(T_{u}T_{v})=\varphi(T_{v}T_{u})$ for all $u,v\in W_{m}$. If $u=\varnothing$, then $T_{u}=I$ by Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §word-products and both sides equal $\varphi(T_{v})$; likewise if $v=\varnothing$. For nonempty $u,v$, $T_{u}T_{v}=T_{uv}$ and $T_{v}T_{u}=T_{vu}$ 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

$$\varphi(T_{uv})=\varphi(T_{vu})\qquad\text{for all nonempty }u,v\in W_{m}.\qquad(3.1)$$

First, for a letter $(i)$ and a nonempty word $v$ of length $l$ with letters $v_{1},\dots,v_{l}$, we have $\varphi(T_{(i)v})=\varphi(T_{v(i)})$ (3.2). Indeed, by Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §word-products and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, $T_{(i)v}=T_{i}T_{v_{1}}\cdots T_{v_{l}}$ and $T_{v(i)}=T_{v_{1}}\cdots T_{v_{l}}T_{i}$, products of the same $l+1\ge2$ factors $\Lambda_{f(\cdot)}(c_{\cdot})$ with the first moved to the end, so (3.2) 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 (3.1) by induction on the length $l'\in\mathbb{N}$ of $u$, for all nonempty $v$ simultaneously. If $l'=1$, then $u$ is a letter by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter and (3.1) is (3.2). Suppose (3.1) holds for all $u$ of length $l'$, and let $u$ have length $l'+1$. By Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter, $u=u'(i)$ with $u'$ of length $l'$; by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §monoid, $uv=u'((i)v)$, $((i)v)u'=(i)(vu')$ and $(vu')(i)=vu$, the words $(i)v$ and $vu'$ being nonempty. The induction hypothesis (for $u'$ and $(i)v$) and (3.2) (for $i$ and $vu'$) give

$$\varphi(T_{uv})=\varphi(T_{u'((i)v)})=\varphi(T_{(i)(vu')})=\varphi(T_{(vu')(i)})=\varphi(T_{vu}).$$

(c) The law. 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 (a), (b) and The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State §law (with $m$, $\mathcal{H}$, $T$ in place of its $n$, $H$, $T$), the map

$$\Gamma':\mathcal{P}_{m}\to\mathbb{C},\qquad\Gamma'(p)=\langle\Omega,p(T)\Omega\rangle_{\mathcal{H}}=\varphi(p(T)),$$

belongs to $\Sigma_{(k+2)d,R}$. It is determined by $(k,\Gamma)$.

Step 4 (one gluing step: the marginals of $\Gamma'$). (a) Let $p\in\mathcal{P}_{(k+1)d}$. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution (for the tuple $(x_{1},\dots,x_{(k+1)d})$ of $\beta_{k}$ and the tuple $T$) and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, $(\beta_{k}p)(T)=p(T_{1},\dots,T_{(k+1)d})=p(\Lambda_{1}(L^{\Gamma}_{x_{1}}),\dots,\Lambda_{1}(L^{\Gamma}_{x_{(k+1)d}}))$. Apply Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §transport with $\mathcal{H}_{\Gamma}$, $\mathcal{H}$, $(k+1)d$, $(L^{\Gamma}_{x_{1}},\dots,L^{\Gamma}_{x_{(k+1)d}})$, $A_{1}$, $\Lambda_{1}$ in place of its $H$, $K$, $n$, $T$, $\mathcal{A}$, $\Phi$: the set $A_{1}$ contains $I$ and these operators 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. With Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §gns (for $\Gamma\in\Sigma_{(k+1)d}$), which gives $p(L^{\Gamma}_{x_{1}},\dots,L^{\Gamma}_{x_{(k+1)d}})=L^{\Gamma}_{p}$, this yields $(\beta_{k}p)(T)=\Lambda_{1}(L^{\Gamma}_{p})$. 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, $\Gamma'(\beta_{k}p)=\varphi(\Lambda_{1}(L^{\Gamma}_{p}))=\tau_{\Gamma}(L^{\Gamma}_{p})=\Gamma(p)$. Thus $\Gamma'\circ\beta_{k}=\Gamma$.

(b) Let $q\in\mathcal{P}_{2d}$. The tuple of $\delta_{k+1}$ is $(x_{1},\dots,x_{d},x_{(k+1)d+1},\dots,x_{(k+2)d})$, so as in (a), $(\delta_{k+1}q)(T)=q(T_{1},\dots,T_{d},T_{(k+1)d+1},\dots,T_{(k+2)d})$. By (2.1), $T_{j}=\Lambda_{1}(L^{\Gamma}_{x_{j}})=\Lambda_{2}(L^{\gamma_{k+1}}_{x_{j}})$, and by definition $T_{(k+1)d+j}=\Lambda_{2}(L^{\gamma_{k+1}}_{x_{d+j}})$, for $j\in[d]$. So this $2d$-tuple is $(\Lambda_{2}(L^{\gamma_{k+1}}_{x_{1}}),\dots,\Lambda_{2}(L^{\gamma_{k+1}}_{x_{2d}}))$, and exactly as in (a), with $\mathcal{H}_{\gamma_{k+1}}$, $2d$, $A_{2}$, $\Lambda_{2}$, $\gamma_{k+1}$ in place of $\mathcal{H}_{\Gamma}$, $(k+1)d$, $A_{1}$, $\Lambda_{1}$, $\Gamma$,

$$\Gamma'(\delta_{k+1}q)=\varphi(\Lambda_{2}(L^{\gamma_{k+1}}_{q}))=\tau_{\gamma_{k+1}}(L^{\gamma_{k+1}}_{q})=\gamma_{k+1}(q).$$

Thus $\Gamma'\circ\delta_{k+1}=\gamma_{k+1}$, and with Step 3(c), $\Gamma'$ satisfies $(B_{k+1})$.

Step 5 (the consistent laws, by recursion on $k\in\mathbb{N}$). Let $X=\bigcup_{n\in\mathbb{N}}\Sigma_{n}$ and define $F:\mathbb{N}\times X\to X$ by $F(k,\Gamma)=\Gamma'$, the law of Step 3(c) built from $(k,\Gamma)$, if $\Gamma$ satisfies $(B_{k})$, and $F(k,\Gamma)=\Gamma$ otherwise; this is a map into $X$ since $\Gamma'\in\Sigma_{(k+2)d,R}\subseteq\Sigma_{(k+2)d}$ (Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law) and $\Gamma'$ is determined by $(k,\Gamma)$, so no choice is involved. As $\gamma_{1}\in\Sigma_{2d}\subseteq X$ by Step 1, Definition of Sequences by Recursion on the Natural Numbers §recursion gives a sequence $(\Gamma_{k})_{k\in\mathbb{N}}$ in $X$ with $\Gamma_{1}=\gamma_{1}$ and $\Gamma_{k+1}=F(k,\Gamma_{k})$ for every $k\in\mathbb{N}$. By induction on $k\in\mathbb{N}$, $\Gamma_{k}$ satisfies $(B_{k})$: $\gamma_{1}\in\Sigma_{2d,R}$ by Step 1, and $\delta_{1}=\sigma_{(x_{1},\dots,x_{2d})}$ is the identity of $\mathcal{P}_{2d}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §identity, so $\Gamma_{1}\circ\delta_{1}=\gamma_{1}$; and if $\Gamma_{k}$ satisfies $(B_{k})$, then $\Gamma_{k+1}$ is the law $\Gamma'$ built from $(k,\Gamma_{k})$, which satisfies $(B_{k+1})$ by Step 4(b). Consequently, for every $k\in\mathbb{N}$, Step 4(a) applies to $(k,\Gamma_{k})$, and

$$\Gamma_{k}\in\Sigma_{(k+1)d,R},\qquad\Gamma_{k}\circ\delta_{k}=\gamma_{k},\qquad\Gamma_{k+1}\circ\beta_{k}=\Gamma_{k}.\qquad(5.1)$$

Step 6 (a chain of tracial W-probability spaces).* For $k\in\mathbb{N}$ write $H_{k}=\mathcal{H}_{\Gamma_{k}}$, $M_{k}=\mathcal{M}_{\Gamma_{k}}$, $\Omega_{k}=\Omega_{\Gamma_{k}}$ and $L^{(k)}_{p}=L^{\Gamma_{k}}_{p}$ for $p\in\mathcal{P}_{(k+1)d}$. As $\Gamma_{k}\in\Sigma_{(k+1)d}$, The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star shows that $(H_{k},M_{k},\Omega_{k})$ is a tracial W*-probability space with trace $\tau_{\Gamma_{k}}$. Let $\theta_{k}:M_{k}\to M_{k+1}$ 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 $(k+2)d$, $(k+1)d$, $\Gamma_{k+1}$ and the tuple $(x_{1},\dots,x_{(k+1)d})$ in $\mathcal{P}_{(k+2)d,\mathrm{sa}}$ in place of its $m$, $n$, $\gamma$ and $a$; then its $\sigma_{a}$ is $\beta_{k}$, and its marginal law $\gamma\circ\sigma_{a}$ is $\Gamma_{k+1}\circ\beta_{k}=\Gamma_{k}$ by (5.1), so $\theta_{k}$ indeed maps $\mathcal{M}_{\Gamma_{k}}=M_{k}$ into $\mathcal{M}_{\Gamma_{k+1}}=M_{k+1}$. By Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §homomorphism, $\theta_{k}$ is linear, $\theta_{k}(I)=I$, $\theta_{k}(ST)=\theta_{k}(S)\theta_{k}(T)$, $\theta_{k}(T^{*})=\theta_{k}(T)^{*}$ and $\tau_{\Gamma_{k+1}}(\theta_{k}(T))=\tau_{\Gamma_{k}}(T)$ for $S,T\in M_{k}$; so, the traces being $\tau_{\Gamma_{k+1}}$ and $\tau_{\Gamma_{k}}$, $\theta_{k}$ 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

$$\theta_{k}(L^{(k)}_{x_{i}})=L^{(k+1)}_{\beta_{k}(x_{i})}=L^{(k+1)}_{x_{i}}\qquad(i\in[(k+1)d]).\qquad(6.1)$$

Step 7 (the inductive limit and the tuples). Apply Inductive Limits of Tracial W*-Probability Spaces along Trace-Preserving Embeddings with $\theta_{k}$ in place of its $\pi_{k}$: this gives 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 $\rho_{k+1}(\theta_{k}(S))=\rho_{k}(S)$ for $k\in\mathbb{N}$ and $S\in M_{k}$ (Inductive Limits of Tracial W*-Probability Spaces along Trace-Preserving Embeddings §compatible). Let $k\in\mathbb{N}$ and $i\in[(k+1)d]$; we show by induction on $n\in\mathbb{N}$ with $n\ge k$ that

$$\rho_{n}(L^{(n)}_{x_{i}})=\rho_{k}(L^{(k)}_{x_{i}}).\qquad(7.1)$$

For $n=k$ this is trivial. If it holds for $n$, then $i\le(k+1)d\le(n+1)d$, so by (6.1) and compatibility $\rho_{n+1}(L^{(n+1)}_{x_{i}})=\rho_{n+1}(\theta_{n}(L^{(n)}_{x_{i}}))=\rho_{n}(L^{(n)}_{x_{i}})=\rho_{k}(L^{(k)}_{x_{i}})$. Define, for $j\in[d]$ and $k\in\mathbb{N}$,

$$s_{j}=\rho_{1}(L^{(1)}_{x_{j}}),\qquad t^{k}_{j}=\rho_{k}(L^{(k)}_{x_{kd+j}}),$$

and $s=(s_{1},\dots,s_{d})$, $t^{k}=(t^{k}_{1},\dots,t^{k}_{d})$; here $kd+j\in[(k+1)d]$. These are elements of $M$. By Trace-Preserving Embeddings Preserve Operator Norms and Compose §norm 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 §multiplication (for $\Gamma_{k}\in\Sigma_{(k+1)d,R}$, by (5.1)), $\lVert s_{j}\rVert_{\mathrm{op}}=\lVert L^{(1)}_{x_{j}}\rVert_{\mathrm{op}}\le R$ and $\lVert t^{k}_{j}\rVert_{\mathrm{op}}=\lVert L^{(k)}_{x_{kd+j}}\rVert_{\mathrm{op}}\le R$.

Step 8 (the laws). Fix $k\in\mathbb{N}$ and let $u^{k}$ be the $2d$-tuple $(L^{(k)}_{x_{1}},\dots,L^{(k)}_{x_{d}},L^{(k)}_{x_{kd+1}},\dots,L^{(k)}_{x_{(k+1)d}})$. Its entries lie 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 are self-adjoint by Notation, so $u^{k}$ is a self-adjoint $2d$-tuple in $M_{k}$ (Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §tuple). By (7.1) (with $1$, $j$, $k$ in place of $k$, $i$, $n$), $s_{j}=\rho_{k}(L^{(k)}_{x_{j}})$ for $j\in[d]$; hence $(\rho_{k}(u^{k}_{1}),\dots,\rho_{k}(u^{k}_{2d}))=(s_{1},\dots,s_{d},t^{k}_{1},\dots,t^{k}_{d})$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §embedding (with $2d$, $(H_{k},M_{k},\Omega_{k})$, $(H,M,\Omega)$, $\rho_{k}$, $u^{k}$ in place of its $d$, $(H,M,\Omega)$, $(K,N,\Psi)$, $\pi$, $s$), this is a self-adjoint $2d$-tuple in $M$ with law $\lambda_{u^{k}}$. In particular every $s_{j}$ and $t^{k}_{j}$ is a self-adjoint element of $M$, so $s$ and $t^{k}$ are self-adjoint $d$-tuples in $M$, and the $2d$-tuple $(s,t^{k})$ of Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling is this tuple; thus $\lambda_{(s,t^{k})}=\lambda_{u^{k}}$.

Let $e$ be the $2d$-tuple $(x_{1},\dots,x_{d},x_{kd+1},\dots,x_{(k+1)d})$ of $\delta_{k}$ and $L^{(k)}_{x}=(L^{(k)}_{x_{1}},\dots,L^{(k)}_{x_{(k+1)d}})$. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, $e(L^{(k)}_{x})=u^{k}$, so for $p\in\mathcal{P}_{2d}$, Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law, Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution, The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §law (for $\Gamma_{k}\in\Sigma_{(k+1)d}$) and (5.1) give

$$\lambda_{u^{k}}(p)=\langle\Omega_{k},p(u^{k})\Omega_{k}\rangle=\langle\Omega_{k},(\delta_{k}p)(L^{(k)}_{x})\Omega_{k}\rangle=\Gamma_{k}(\delta_{k}p)=\gamma_{k}(p).$$

Hence $\lambda_{(s,t^{k})}=\gamma_{k}$. By Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling, $\lambda_{(s,t^{k})}\in\Pi(\lambda_{s},\lambda_{t^{k}})$, so by Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling, applied to this coupling and to $\gamma_{k}\in\Pi(\mu,\nu_{k})$,

$$\lambda_{s}=\lambda_{(s,t^{k})}\circ\iota^{1}=\gamma_{k}\circ\iota^{1}=\mu,\qquad\lambda_{t^{k}}=\lambda_{(s,t^{k})}\circ\iota^{2}=\gamma_{k}\circ\iota^{2}=\nu_{k}.$$

Together with the norm bounds of Step 7, this proves the claim.