Proof of Gluing a Chain of Noncommutative Couplings into Consistent Joint Laws

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

Notation. For a law $\lambda$ and a polynomial $p$ in its variables, $L^{\lambda}_{p}$ denotes 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; the superscript names the GNS space, since three of them occur together below. The amalgamated free product space of Step 2 is written $\mathcal{H}$ without subscript, as in The Amalgamated Free Product Space of Two Noncommutative Laws over a Common Marginal, Its Vacuum Vector and Vacuum State §space; GNS spaces always carry a subscript. The letter $\varepsilon$ is reserved for the substitutions $\varepsilon_{k}$ of the statement: the two factors of an amalgamated free product are indexed by the explicit numbers $1$ and $2$, not by the index letter of Two Noncommutative Laws with a Common Marginal: Standing Notation for Their Amalgamated Free Product §data. $I$ is the identity operator; the cost $I(\gamma)$ is not used. For $k\in\mathbb{N}$, a map $\Gamma$ satisfies $(B_{k})$ if $\Gamma\in\Sigma_{kd}$ and $\Gamma\circ\varepsilon_{k}=\lambda_{k}$. Every variable is self-adjoint: the support of $x_{j}$ is the one-letter word $(j)$, which is its own reversal, so $x_{j}^{*}=x_{j}$ by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §adjoint, that is $x_{j}$ is self-adjoint in the sense of The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §self-adjoint.

Step 1. The couplings are laws. Let $k\in\mathbb{N}$. By Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law there are reals $r,r'>0$ with $\lambda_{k}\in\Sigma_{d,r}$ and $\lambda_{k+1}\in\Sigma_{d,r'}$; for $R_{k}=\max(r,r')$ both lie in $\Sigma_{d,R_{k}}$ by Basic Properties of Noncommutative Laws: Adjoints, the Self-Adjoint Pairing, Cauchy-Schwarz, Monotonicity in the Bound, and Laws of Constant Tuples §monotone. Hence $\gamma_{k}\in\Sigma_{2d,R_{k}}$ by Couplings of Noncommutative Laws: the Norm Bound, the Cost Identity, the Tensor, Diagonal and Swapped Couplings, Weak-Star Closedness, and Displacement Interpolants §bound, and so $\gamma_{k}\in\Sigma_{2d}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law.

Step 2. The amalgamation data of one step. 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 $m_{1}=kd$, $\gamma_{1}=\Gamma$ and the $d$-tuple $a^{1}=(x_{(k-1)d+1},\dots,x_{kd})$ in $\mathcal{P}_{kd,\mathrm{sa}}$, and with $m_{2}=2d$, $\gamma_{2}=\gamma_{k}$ and the $d$-tuple $a^{2}=(x_{1},\dots,x_{d})$ in $\mathcal{P}_{2d,\mathrm{sa}}$. Here $\Gamma\in\Sigma_{kd}$ by $(B_{k})$ and $\gamma_{k}\in\Sigma_{2d}$ by Step 1. The common marginal condition holds with marginal $\lambda_{k}$: $\sigma_{a^{1}}=\varepsilon_{k}$ by the definition of $\varepsilon_{k}$, so $\Gamma\circ\sigma_{a^{1}}=\lambda_{k}$ by $(B_{k})$; and $\sigma_{a^{2}}=\iota^{1}$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals, so $\gamma_{k}\circ\sigma_{a^{2}}=\lambda_{k}$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling. Thus, 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, $N=\mathcal{M}_{\lambda_{k}}$, $A_{1}=\mathcal{M}_{\Gamma}$, $A_{2}=\mathcal{M}_{\gamma_{k}}$, 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 $(\gamma_{k},a^{2})$. 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), 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 $j\in[d]$, Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §homomorphism gives $\pi_{1}(L^{\lambda_{k}}_{x_{j}})=L^{\Gamma}_{\sigma_{a^{1}}(x_{j})}$ and $\pi_{2}(L^{\lambda_{k}}_{x_{j}})=L^{\gamma_{k}}_{\sigma_{a^{2}}(x_{j})}$, and $\sigma_{a^{1}}(x_{j})=x_{(k-1)d+j}$, $\sigma_{a^{2}}(x_{j})=x_{j}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values. Hence

Step 3. The operator tuple and its law. With the data of Step 2, let $T$ be the $(k+1)d$-tuple in $\mathcal{L}(\mathcal{H})$ with

The operators $L^{\Gamma}_{x_{i}}$ lie in $A_{1}$ and the operators $L^{\gamma_{k}}_{x_{d+j}}$ in $A_{2}$ 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, so every $T_{i}$ has the form $\Lambda_{f}(c)$ with $f\in\{1,2\}$ and $c\in A_{f}$.

(a) Self-adjointness. Each such $c$ is $L_{x_{l}}$ on the relevant GNS space, and $c^{*}=L_{x_{l}^{*}}=c$ 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. By The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §representation, $T_{i}^{*}=\Lambda_{f}(c)^{*}=\Lambda_{f}(c^{*})=T_{i}$, so by Adjoint of a Linear Map between Complex Inner Product Spaces §adjoint $\langle T_{i}\xi,\eta\rangle=\langle\xi,T_{i}\eta\rangle$ for all $\xi,\eta\in\mathcal{H}$; that is, $T_{i}$ is self-adjoint in the sense of Self-Adjoint Operator.

(b) Norm bound. By Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law fix reals $r_{1},r_{2}>0$ with $\Gamma\in\Sigma_{kd,r_{1}}$ and $\gamma_{k}\in\Sigma_{2d,r_{2}}$, and put $R=\max(r_{1},r_{2})$. By The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §representation 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, $\lVert T_{i}\rVert_{\mathrm{op}}\le\lVert L^{\Gamma}_{x_{i}}\rVert_{\mathrm{op}}\le r_{1}\le R$ for $i\in[kd]$ and $\lVert T_{kd+j}\rVert_{\mathrm{op}}\le\lVert L^{\gamma_{k}}_{x_{d+j}}\rVert_{\mathrm{op}}\le r_{2}\le R$ for $j\in[d]$.

(c) Traciality. 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 The Amalgamated Free Product Space of Two Noncommutative Laws over a Common Marginal, Its Vacuum Vector and Vacuum State §state, $\langle\Omega,A\Omega\rangle=\varphi(A)$ for $A\in\mathcal{L}(\mathcal{H})$, and by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, $T_{u}T_{v}=T_{uv}$ and $T_{v}T_{u}=T_{vu}$ for $u,v\in W_{(k+1)d}$. So it suffices to show $\varphi(T_{uv})=\varphi(T_{vu})$ for all words $u,v$. If $u$ or $v$ is empty, then $uv=vu$. Rotation: let $w=(i)w'$ with $i\in[(k+1)d]$ and $w'$ a nonempty word, of total length $r\ge2$. By Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §word-products, $T_{w}$ and $T_{w'(i)}$ are the products $T_{w_{1}}\cdots T_{w_{r}}$ and $T_{w_{2}}\cdots T_{w_{r}}T_{w_{1}}$ of $r$ factors of the form $\Lambda_{f}(c)$ with $c\in A_{f}$, so The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §trace gives $\varphi(T_{(i)w'})=\varphi(T_{w'(i)})$. Now we show, by induction on the length $l\ge1$ of $u$, that $\varphi(T_{uv})=\varphi(T_{vu})$ for every nonempty $v$. For $l=1$, $u=(i)$ and this is the rotation with $w'=v$. If it holds for length $l$ and $u=(i)u'$ with $u'$ of length $l$, then, by the rotation with $w'=u'v$ and the induction hypothesis for $u'$ and the nonempty word $v(i)$,

(d) Conclusion. By (a), (b), (c) and The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State §law (with $n=(k+1)d$ and the Hilbert space $\mathcal{H}$), the map

belongs to $\Sigma_{(k+1)d,R}$, hence to $\Sigma_{(k+1)d}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law. It is determined by $(k,\Gamma)$; $R$ enters only this membership proof.

Step 4. The marginals of $\Gamma'$. Keep the data of Steps 2 and 3.

(a) Let $p\in\mathcal{P}_{kd}$. By Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution, applied to the $kd$-tuple $(x_{1},\dots,x_{kd})$ of $\beta_{k}$ and the tuple $T$, and by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, $(\beta_{k}p)(T)=p(T_{1},\dots,T_{kd})$. Apply Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §transport to the $kd$-tuple $(L^{\Gamma}_{x_{1}},\dots,L^{\Gamma}_{x_{kd}})$ in $\mathcal{L}(\mathcal{H}_{\Gamma})$, the set $A_{1}=\mathcal{M}_{\Gamma}$ (which 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), $K=\mathcal{H}$ and $\Phi=\Lambda_{1}$ (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 $\Gamma$ this gives

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 $q\in\mathcal{P}_{2d}$, and let $b=(x_{(k-1)d+1},\dots,x_{(k+1)d})$ be the $2d$-tuple of $\delta_{k}$. By 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, $(\delta_{k}q)(T)=q(T_{(k-1)d+1},\dots,T_{(k+1)d})$. For $j\in[d]$ the index $(k-1)d+j$ lies in $[kd]$, and by (2.1), by The Amalgamated Free Product of Two Noncommutative Laws over a Common Marginal: Representations, Amalgamation, Freeness and the Tracial Vacuum State §amalgamation (applicable because $L^{\lambda_{k}}_{x_{j}}\in N$ 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 by (2.1) again,

while $T_{kd+j}=\Lambda_{2}(L^{\gamma_{k}}_{x_{d+j}})$ by definition. So the $2d$-tuple $(T_{(k-1)d+1},\dots,T_{(k+1)d})$ is $(\Lambda_{2}(L^{\gamma_{k}}_{x_{1}}),\dots,\Lambda_{2}(L^{\gamma_{k}}_{x_{2d}}))$, and exactly as in (a), with $A_{2}=\mathcal{M}_{\gamma_{k}}$, $\Phi=\Lambda_{2}$ and $\gamma_{k}$ in place of $A_{1}$, $\Lambda_{1}$ and $\Gamma$,

Thus $\Gamma'\circ\delta_{k}=\gamma_{k}$.

(c) By Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals, $\iota^{2}=\sigma_{(x_{d+1},\dots,x_{2d})}$. By Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §composition, $\delta_{k}\circ\iota^{2}=\sigma_{c}$ with $c_{j}=\delta_{k}(x_{d+j})$, and $\delta_{k}(x_{d+j})=b_{d+j}=x_{(k-1)d+d+j}=x_{kd+j}$ by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §values (for $k=1$ both sides are $x_{d+j}$ by the convention of the statement). Since $(k+1-1)d+j=kd+j$, $c$ is the tuple of $\varepsilon_{k+1}$, so $\delta_{k}\circ\iota^{2}=\varepsilon_{k+1}$. By (b) and Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling,

With Step 3(d), $\Gamma'$ satisfies $(B_{k+1})$.

Step 5. The recursion. Let $X=\bigcup_{n\in\mathbb{N}}\Sigma_{nd}$ and define $F:\mathbb{N}\times X\to X$ by $F(k,\Gamma)=\Gamma'$, the law of Step 3(d) built from $(k,\Gamma)$, if $\Gamma$ satisfies $(B_{k})$, and $F(k,\Gamma)=\Gamma$ otherwise. This is a map into $X$, since $\Gamma'\in\Sigma_{(k+1)d}$ and $\Gamma'$ is determined by $(k,\Gamma)$ (Step 2); no choice is involved. As $\lambda_{1}\in\Sigma_{d}\subseteq X$, Definition of Sequences by Recursion on the Natural Numbers §recursion gives a sequence $(\Gamma_{k})_{k\in\mathbb{N}}$ in $X$ with $\Gamma_{1}=\lambda_{1}$ and $\Gamma_{k+1}=F(k,\Gamma_{k})$ for every $k\in\mathbb{N}$. Clause 1 holds.

Step 6. Verification. We show by induction that $\Gamma_{k}$ satisfies $(B_{k})$ for every $k\in\mathbb{N}$. For $k=1$: $\Gamma_{1}=\lambda_{1}\in\Sigma_{d}$, and $\varepsilon_{1}=\sigma_{(x_{1},\dots,x_{d})}:\mathcal{P}_{d}\to\mathcal{P}_{d}$ is the identity map by Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §identity, so $\Gamma_{1}\circ\varepsilon_{1}=\lambda_{1}$. If $\Gamma_{k}$ satisfies $(B_{k})$, then $\Gamma_{k+1}=F(k,\Gamma_{k})$ is the law $\Gamma'$ built from $(k,\Gamma_{k})$, which satisfies $(B_{k+1})$ by Step 4(c).

Consequently $\Gamma_{k}\in\Sigma_{kd}$ and $\Gamma_{k}\circ\varepsilon_{k}=\lambda_{k}$ for every $k$, which is clause 4. Moreover, for every $k$, $\Gamma_{k+1}$ is the law $\Gamma'$ built from $(k,\Gamma_{k})$, so Step 4(a) gives $\Gamma_{k+1}\circ\beta_{k}=\Gamma_{k}$ (clause 2) and Step 4(b) gives $\Gamma_{k+1}\circ\delta_{k}=\gamma_{k}$ (clause 3).