Proof of Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum

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

Conventions. (C1) Evaluation. Let $(K,N,\Psi)$ be a tracial W*-probability space and $T$ a tuple in $N$. For a polynomial $p$, $p(T)$ is the value of Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §evaluation; it is linear in $p$, $(pq)(T)=p(T)q(T)$, $1(T)=I$, $x_{j}(T)=T_{j}$ and $T_{uv}=T_{u}T_{v}$ for words $u,v$, $p^{*}(T)=p(T)^{*}$ if the entries of $T$ are self-adjoint, and $(\sigma_{a}(p))(S)=p(a(S))$ for a tuple $a$ of polynomials with substitution $\sigma_{a}$, by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values, Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §homomorphism, Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §adjoint and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution. Moreover $p(T)\in N$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §transport (with $\Phi$ the identity map of $N$), and $p(T)$ is self-adjoint if the entries of $T$ are self-adjoint and $p$ is a self-adjoint polynomial. For a self-adjoint $n$-tuple $z$ in $N$, $\lambda_{z}(p)=\langle\Psi,p(z)\Psi\rangle$ by Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law, so for $q\in\mathcal{P}_{n}$

$$\lVert q(z)\Psi\rVert^{2}=\langle\Psi,q(z)^{*}q(z)\Psi\rangle=\lambda_{z}(q^{*}q).\tag{E}$$

(C2) Law invariance. If $Z,Z'$ are $L^{2}$ $k$-tuples, possibly of different tracial W*-probability spaces, with $\mathrm{law}(Z)=\mathrm{law}(Z')$, then $\mathrm{law}(TZ)=T_{\#}\mathrm{law}(Z)=\mathrm{law}(TZ')$ for every affine datum $T$ from $k$ variables, by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, and $\langle Z_{i},Z_{l}\rangle=\mathrm{m}_{il}(\mathrm{law}(Z))=\langle Z'_{i},Z'_{l}\rangle$ for all $i,l\in[k]$, by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §moments; in particular $\lVert Z\rVert_{2}=\lVert Z'\rVert_{2}$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples).

(C3) Bounded realisations. Let $(K,N,\Psi)$ be a tracial W*-probability space, $k\in\mathbb{N}$, and $Z$ an $L^{2}$ $k$-tuple of it with $\mathrm{law}(Z)=\kappa_{k}(\gamma)$ for some $\gamma\in\Sigma_{k,\rho}$, $\rho>0$ real. By Square-Integrable Tuples with a Bounded Law are Vacuum Tuples of Bounded Self-Adjoint Operators §operator, Square-Integrable Tuples with a Bounded Law are Vacuum Tuples of Bounded Self-Adjoint Operators §bound and Square-Integrable Tuples with a Bounded Law are Vacuum Tuples of Bounded Self-Adjoint Operators §law there is exactly one self-adjoint $k$-tuple $z$ in $N$ with $z\Psi=Z$, and it satisfies $\lVert z_{j}\rVert_{\mathrm{op}}\le\rho$ for all $j$ and $\lambda_{z}=\gamma$. Every law in $\Sigma_{k}$ lies in some $\Sigma_{k,\rho}$ (Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law). Conversely, for a self-adjoint $k$-tuple $z$ in $N$, $\mathrm{law}(z\Psi)=\kappa_{k}(\lambda_{z})$ by Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded; and for self-adjoint $d$-tuples $z,z'$ in $N$, the $2d$-tuple $(z,z')$ is self-adjoint with vacuum tuple $(z\Psi,z'\Psi)$ and $\lambda_{(z,z')}\in\Pi(\lambda_{z},\lambda_{z'})$ by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling, so that $\lambda_{(z,z')}\circ\iota^{1}=\lambda_{z}$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling.

(C4) GNS tuples. Let $k\in\{d,2d\}$ and $\gamma\in\Sigma_{k}$. Each multiplication operator $L_{x_{i}}$ ($i\in[k]$, 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) lies in $\mathcal{A}_{\gamma}\subseteq\mathcal{A}_{\gamma}''=\mathcal{M}_{\gamma}$ (The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star, The Commutant of a Set of Bounded Operators on a Complex Hilbert Space §commutant) and 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, since $x_{i}^{*}=x_{i}$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint). So $\ell^{\gamma}=(L_{x_{1}},\dots,L_{x_{k}})$ is a self-adjoint $k$-tuple in $\mathcal{M}_{\gamma}$, and $p(\ell^{\gamma})=L_{p}$ for $p\in\mathcal{P}_{k}$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §gns; hence $p(\ell^{\gamma})\Omega_{\gamma}=\widehat{p}$ and $\langle\widehat{p},\widehat{q}\rangle=\gamma(p^{*}q)$ for $p,q\in\mathcal{P}_{k}$ (Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §vacuum), and $\lambda_{\ell^{\gamma}}=\gamma$ by The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §law. In particular $\ell^{\lambda}\Omega_{\lambda}=X_{\lambda}$, and $\ell^{\pi}\Omega_{\pi}=(X_{\pi},P_{\pi})$ for a bounded plan $\pi$ at $\lambda$ (The Shift of a Bounded Plan by a Self-Adjoint Field §tuples).

Step 1 (the operator tuple of $X$). Since $\mathrm{law}(X)=\kappa_{d}(\lambda)$ with $\lambda\in\Sigma_{d,r}$, (C3) gives exactly one self-adjoint $d$-tuple $s$ in $M$ with $s\Omega=X$, and $\lambda_{s}=\lambda$. Put $\ell=\ell^{\lambda}$; by (C4), $\ell\Omega_{\lambda}=X_{\lambda}$, $\lambda_{\ell}=\lambda$ and $q(\ell)\Omega_{\lambda}=\widehat{q}$ for $q\in\mathcal{P}_{d}$.

Step 2 (mixed moments depend only on the joint law). We claim: let $(H_{i},M_{i},\Omega_{i})$, $i\in\{1,2\}$, be tracial W*-probability spaces, $s^{i}$ a self-adjoint $d$-tuple in $M_{i}$ with $\lambda_{s^{i}}=\lambda$, and $Y^{i}$ an $L^{2}$ $d$-tuple of $(H_{i},M_{i},\Omega_{i})$, and suppose $\mathrm{law}(s^{1}\Omega_{1},Y^{1})=\mathrm{law}(s^{2}\Omega_{2},Y^{2})$. Then $\langle Y^{1}_{j},q(s^{1})\Omega_{1}\rangle=\langle Y^{2}_{j},q(s^{2})\Omega_{2}\rangle$ for every $q\in\mathcal{P}_{d}$ and $j\in[d]$.

(a) A Lipschitz bound. Let $(K,N,\Psi)$ be a tracial W*-probability space with conjugation $J_{N}$. For $b\in N$ and self-adjoint $c\in N$, The Commutator of a Square-Integrable Vector with a Bounded Self-Adjoint Operator: Linearity, the Norm Bound, Vacuum Vectors and Dependence on the Law Only §commutator and The Commutator of a Square-Integrable Vector with a Bounded Self-Adjoint Operator: Linearity, the Norm Bound, Vacuum Vectors and Dependence on the Law Only §vacuum give $J_{N}cJ_{N}(b\Psi)-cb\Psi=(bc-cb)\Psi$, that is, $bc\Psi=J_{N}cJ_{N}(b\Psi)$; as $J_{N}$ preserves norms (Conjugation of a Complex Hilbert Space §conjugation), $\lVert bc\Psi\rVert\le\lVert c\rVert_{\mathrm{op}}\lVert b\Psi\rVert$. By induction on $m$, applying this to $bc_{1}\cdots c_{m-1}\in N$ and $c_{m}$, we get $\lVert bc_{1}\cdots c_{m}\Psi\rVert\le r^{m}\lVert b\Psi\rVert$ for $b\in N$ and self-adjoint $c_{1},\dots,c_{m}\in N$ of operator norm at most $r$. Now let $a,a'$ be self-adjoint $d$-tuples in $N$ with $\lVert a_{l}\rVert_{\mathrm{op}}\le r$ and $\lVert a'_{l}\rVert_{\mathrm{op}}\le r$ for all $l\in[d]$, and let $w=i_{1}\cdots i_{n}$ be a word of length $n\ge1$. By (C1), $a_{w}-a'_{w}$ is the sum over $u\in[n]$ of $a_{i_{1}}\cdots a_{i_{u-1}}(a_{i_{u}}-a'_{i_{u}})a'_{i_{u+1}}\cdots a'_{i_{n}}$; the left factor ($I$ for $u=1$, the value at the empty word) has operator norm at most $r^{u-1}$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §bound, and the bound just proved, with $b=a_{i_{u}}-a'_{i_{u}}$, bounds the norm of the rest applied to $\Psi$ by $r^{n-u}\lVert a_{i_{u}}\Psi-a'_{i_{u}}\Psi\rVert$. Since $\lVert a_{i}\Psi-a'_{i}\Psi\rVert\le\lVert a\Psi-a'\Psi\rVert_{2}$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples), $\lVert a_{w}\Psi-a'_{w}\Psi\rVert\le nr^{n-1}\lVert a\Psi-a'\Psi\rVert_{2}$, and the difference vanishes for the empty word. Writing $q=\sum_{w}c_{w}x_{w}$ (a finite sum) and $C_{q}=\sum_{w\ne\varnothing}|c_{w}|\,|w|\,r^{|w|-1}$, linearity of evaluation gives

$$\lVert q(a)\Psi-q(a')\Psi\rVert\le C_{q}\lVert a\Psi-a'\Psi\rVert_{2}.\tag{A1}$$

(b) Approximation. The entries of $Y^{i}$ are fixed by the conjugation of $(H_{i},M_{i},\Omega_{i})$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples), so by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §limit we may choose, for $i\in\{1,2\}$ and $k\in\mathbb{N}$, self-adjoint $d$-tuples $y^{i,k}$ in $M_{i}$ with $t_{i,k}=\lVert y^{i,k}\Omega_{i}-Y^{i}\rVert_{2}\to0$ as $k\to\infty$. Put $\gamma^{i,k}=\lambda_{(s^{i},y^{i,k})}$, a law in $\Sigma_{2d}$ by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law; by (C3), $\kappa_{2d}(\gamma^{i,k})=\mathrm{law}(s^{i}\Omega_{i},y^{i,k}\Omega_{i})$ and $\gamma^{i,k}\circ\iota^{1}=\lambda_{s^{i}}=\lambda$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §lipschitz, $\widehat{W}_{2}(\kappa_{2d}(\gamma^{i,k}),\mathrm{law}(s^{i}\Omega_{i},Y^{i}))\le t_{i,k}$; as the laws $\mathrm{law}(s^{1}\Omega_{1},Y^{1})$ and $\mathrm{law}(s^{2}\Omega_{2},Y^{2})$ coincide, the triangle inequality and the symmetry of the metric $\widehat{W}_{2}$ give $e_{k}=\widehat{W}_{2}(\kappa_{2d}(\gamma^{1,k}),\kappa_{2d}(\gamma^{2,k}))\le t_{1,k}+t_{2,k}$. Fix $q\in\mathcal{P}_{d}$ and $j\in[d]$, and let $f=x_{d+j}\,\iota^{1}(q)\in\mathcal{P}_{2d}$. As $\iota^{1}$ is the substitution of $(x_{1},\dots,x_{d})$ (Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals), (C1) gives $f(s^{i},y^{i,k})=y^{i,k}_{j}\,q(s^{i})$, so $\gamma^{i,k}(f)=\langle\Omega_{i},y^{i,k}_{j}q(s^{i})\Omega_{i}\rangle=\langle y^{i,k}_{j}\Omega_{i},q(s^{i})\Omega_{i}\rangle$, $y^{i,k}_{j}$ being self-adjoint. By the Cauchy--Schwarz inequality,

$$\bigl|\langle Y^{i}_{j},q(s^{i})\Omega_{i}\rangle-\gamma^{i,k}(f)\bigr|\le t_{i,k}\,\lVert q(s^{i})\Omega_{i}\rVert\qquad(i\in\{1,2\},\ k\in\mathbb{N}).\tag{A2}$$

(c) Comparison. Fix $k$. By Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs §distance, for $2d$ variables and $\varepsilon=1/k$, there are a tracial W*-probability space $(K,N,\Psi)$ and $L^{2}$ $2d$-tuples $Z,Z'$ of it with $\mathrm{law}(Z)=\kappa_{2d}(\gamma^{1,k})$, $\mathrm{law}(Z')=\kappa_{2d}(\gamma^{2,k})$ and $\lVert Z-Z'\rVert_{2}^{2}\le e_{k}^{2}+1/k$. By (C3) there are self-adjoint $d$-tuples $a,b,a',b'$ in $N$ with $Z=(a\Psi,b\Psi)$, $Z'=(a'\Psi,b'\Psi)$, $\lambda_{(a,b)}=\gamma^{1,k}$ and $\lambda_{(a',b')}=\gamma^{2,k}$; then $\lambda_{a}=\gamma^{1,k}\circ\iota^{1}=\lambda$ and likewise $\lambda_{a'}=\lambda$. So $\mathrm{law}(a\Psi)=\kappa_{d}(\lambda)$ with $\lambda\in\Sigma_{d,r}$, and the uniqueness in (C3) gives $\lVert a_{l}\rVert_{\mathrm{op}}\le r$, and likewise $\lVert a'_{l}\rVert_{\mathrm{op}}\le r$, for all $l$. As in (b), $\gamma^{1,k}(f)=\langle b_{j}\Psi,q(a)\Psi\rangle$ and $\gamma^{2,k}(f)=\langle b'_{j}\Psi,q(a')\Psi\rangle$, and $\lVert q(a)\Psi\rVert^{2}=\lambda(q^{*}q)$ by (E). Hence, by the Cauchy--Schwarz inequality and (A1),

$$\bigl|\gamma^{1,k}(f)-\gamma^{2,k}(f)\bigr|\le\lVert b_{j}\Psi-b'_{j}\Psi\rVert\,\lVert q(a)\Psi\rVert+\lVert b'_{j}\Psi\rVert\,\lVert q(a)\Psi-q(a')\Psi\rVert\le\lVert Z-Z'\rVert_{2}\bigl(\lambda(q^{*}q)^{1/2}+C_{q}\lVert Z'\rVert_{2}\bigr).$$

By (C2), $\lVert Z'\rVert_{2}=\lVert(s^{2}\Omega_{2},y^{2,k}\Omega_{2})\rVert_{2}\le\lVert s^{2}\Omega_{2}\rVert_{2}+\lVert Y^{2}\rVert_{2}+t_{2,k}$, which is bounded in $k$, while $\lVert Z-Z'\rVert_{2}^{2}\le(t_{1,k}+t_{2,k})^{2}+1/k\to0$. So $\gamma^{1,k}(f)-\gamma^{2,k}(f)\to0$; together with (A2), whose right sides tend to $0$ as the vectors $q(s^{i})\Omega_{i}$ do not depend on $k$, the two numbers of the claim differ by less than every positive real, so they are equal.

Step 3 (polynomial approximation of $\zeta$). Each $\zeta_{l}$ is fixed by $J_{\lambda}$, the conjugation of $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples, The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star). Let $k\in\mathbb{N}$ and $l\in[d]$. By The Complex GNS Space of a Tracial State on Noncommutative Polynomials §gns, The Complex GNS Space of a Tracial State on Noncommutative Polynomials §classes and The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §dense there is $p\in\mathcal{P}_{d}$ with $\lVert\widehat{p}-\zeta_{l}\rVert<1/k$. Put $q^{k}_{l}=\frac{1}{2}(p+p^{*})$, an element of $\mathcal{P}_{d,\mathrm{sa}}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint ($p+p^{*}$ is self-adjoint and $\mathcal{P}_{d,\mathrm{sa}}$ is closed under real multiples). The class map is complex-linear (The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §isometry), $J_{\lambda}\widehat{p}=\widehat{p^{*}}$, and $J_{\lambda}$ is additive, conjugate-homogeneous and norm-preserving (Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §conjugation); as $J_{\lambda}\zeta_{l}=\zeta_{l}$,

$$\widehat{q^{k}_{l}}-\zeta_{l}=\tfrac{1}{2}\bigl((\widehat{p}-\zeta_{l})+J_{\lambda}(\widehat{p}-\zeta_{l})\bigr),\qquad\lVert\widehat{q^{k}_{l}}-\zeta_{l}\rVert\le\lVert\widehat{p}-\zeta_{l}\rVert<\tfrac{1}{k}.$$

Write $q^{k}=(q^{k}_{1},\dots,q^{k}_{d})$ and $q^{k}(s)\Omega=(q^{k}_{1}(s)\Omega,\dots,q^{k}_{d}(s)\Omega)$.

Step 4 (the representation). Let $Y$ be an $L^{2}$ $d$-tuple of $(H,M,\Omega)$ with $\mathrm{law}(X,Y)=\mathrm{law}(X_{\lambda},\zeta)$. We claim that

$$\lVert Y_{l}-q(s)\Omega\rVert=\lVert\zeta_{l}-\widehat{q}\rVert\qquad(q\in\mathcal{P}_{d},\ l\in[d]),\tag{R}$$

and hence, by Step 3, $\lVert Y_{l}-q^{k}_{l}(s)\Omega\rVert<1/k$ for all $k$ and $l$, so that $Y_{l}$ is the limit in $H$ of the sequence $(q^{k}_{l}(s)\Omega)_{k\in\mathbb{N}}$. In a complex Hilbert space $\lVert u-v\rVert^{2}=\lVert u\rVert^{2}-2\operatorname{Re}\langle u,v\rangle+\lVert v\rVert^{2}$; we compare the three terms for $(u,v)=(Y_{l},q(s)\Omega)$ and for $(u,v)=(\zeta_{l},\widehat{q})$. First, $\lVert Y_{l}\rVert=\lVert\zeta_{l}\rVert$ by (C2), applied to $(X,Y)$ and $(X_{\lambda},\zeta)$ with both indices equal to $d+l$. Second, $\langle Y_{l},q(s)\Omega\rangle=\langle\zeta_{l},q(\ell)\Omega_{\lambda}\rangle=\langle\zeta_{l},\widehat{q}\rangle$ by Step 2, applied with $s^{1}=s$, $Y^{1}=Y$ in $(H,M,\Omega)$ and $s^{2}=\ell$, $Y^{2}=\zeta$ in $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ (here $\lambda_{s}=\lambda_{\ell}=\lambda$ and $\mathrm{law}(s\Omega,Y)=\mathrm{law}(X,Y)=\mathrm{law}(X_{\lambda},\zeta)=\mathrm{law}(\ell\Omega_{\lambda},\zeta)$ by Step 1), and by Step 1. Third, $\lVert q(s)\Omega\rVert^{2}=\lambda_{s}(q^{*}q)=\lambda(q^{*}q)=\lVert\widehat{q}\rVert^{2}$ by (E) and (C4). This proves (R).

Step 5 (claim 1). If $\mathrm{law}(X,Q)=\mathrm{law}(X,Q')=\mathrm{law}(X_{\lambda},\zeta)$, Step 4 with $Y=Q$ and with $Y=Q'$ shows that $Q_{l}$ and $Q'_{l}$ are both limits of $(q^{k}_{l}(s)\Omega)_{k}$ in the metric space $H$, so $Q_{l}=Q'_{l}$ for every $l\in[d]$ by Uniqueness of Limits in a Metric Space. Hence $Q=Q'$, which is claim 1.

Step 6 (claim 2: the joint law). Assume $\mathrm{law}(X,Q)=\mathrm{law}(X_{\lambda},\zeta)$, and let $\pi$ and $P$ be as in claim 2; thus $\pi\in\Sigma_{2d}$ and $\pi\circ\iota^{1}=\lambda$ (Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plans). By (C3) there are self-adjoint $d$-tuples $x,p$ in $M$ with $(x\Omega,p\Omega)=(X,P)$ and $\lambda_{(x,p)}=\pi$; as $x\Omega=X=s\Omega$, the uniqueness in Step 1 gives $x=s$, so $\lambda_{(s,p)}=\pi$. Let $k\in\mathbb{N}$, and let $\rho^{k}$ be the $3d$-tuple in $\mathcal{P}_{2d}$ with $\rho^{k}_{i}=x_{i}$ for $i\in[2d]$ and $\rho^{k}_{2d+l}=\iota^{1}(q^{k}_{l})$ for $l\in[d]$, with substitution $\sigma_{\rho^{k}}:\mathcal{P}_{3d}\to\mathcal{P}_{2d}$ (Substitution of Noncommutative Polynomials into the Variables §substitution); its entries are self-adjoint by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint and Substitution is the Unique Unital Homomorphism with Prescribed Values on the Variables: Monomials, Products, Adjoints and Composition §adjoint. Let $t^{k}=(s,p,q^{k}_{1}(s),\dots,q^{k}_{d}(s))$, a self-adjoint $3d$-tuple in $M$ by (C1), and $u^{k}=(L_{\rho^{k}_{1}},\dots,L_{\rho^{k}_{3d}})$, a self-adjoint $3d$-tuple in $\mathcal{M}_{\pi}$ by (C4) and (C1). By (C1), $t^{k}$ is the tuple of values of $\rho^{k}$ at $(s,p)$ (as $\iota^{1}$ is the substitution of $(x_{1},\dots,x_{d})$, Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals), and by (C4) $u^{k}$ is the tuple of values of $\rho^{k}$ at $\ell^{\pi}$. Hence, for $f\in\mathcal{P}_{3d}$, $f(t^{k})=(\sigma_{\rho^{k}}f)(s,p)$ and $f(u^{k})=(\sigma_{\rho^{k}}f)(\ell^{\pi})$ by (C1), and by Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law and (C4)

$$\lambda_{t^{k}}(f)=\lambda_{(s,p)}(\sigma_{\rho^{k}}f)=\pi(\sigma_{\rho^{k}}f)=\lambda_{\ell^{\pi}}(\sigma_{\rho^{k}}f)=\lambda_{u^{k}}(f).$$

The vacuum tuple $t^{k}\Omega=(X,P,q^{k}(s)\Omega)$ converges entrywise to $(X,P,Q)$ by Step 4 with $Y=Q$. By (C4), $u^{k}\Omega_{\pi}=(X_{\pi},P_{\pi},\widehat{\iota^{1}(q^{k}_{1})},\dots,\widehat{\iota^{1}(q^{k}_{d})})$, and $\widehat{\iota^{1}(q^{k}_{l})}^{\,\pi}=V^{1}_{\pi}\widehat{q^{k}_{l}}^{\,\lambda}$ by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §isometries, as $\pi\circ\iota^{1}=\lambda$; since $V^{1}_{\pi}$ is linear with $(V^{1}_{\pi})^{*}V^{1}_{\pi}=I$ (Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry), $\lVert V^{1}_{\pi}\widehat{q^{k}_{l}}-V^{1}_{\pi}\zeta_{l}\rVert=\lVert\widehat{q^{k}_{l}}-\zeta_{l}\rVert<1/k$ by Step 3, so $u^{k}\Omega_{\pi}$ converges entrywise to $(X_{\pi},P_{\pi},V^{1}_{\pi}\zeta)$. By Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §law, applied in $(H,M,\Omega)$ to $(t^{k})_{k}$ and in $(\mathcal{H}_{\pi},\mathcal{M}_{\pi},\Omega_{\pi})$ to $(u^{k})_{k}$, both $\mathrm{law}(X,P,Q)$ and $\mathrm{law}(X_{\pi},P_{\pi},V^{1}_{\pi}\zeta)$ are limits of the sequence $(\kappa_{3d}(\lambda_{t^{k}}))_{k}=(\kappa_{3d}(\lambda_{u^{k}}))_{k}$ in $(\Sigma^{2}_{3d},\widehat{W}_{2})$, so they are equal by Uniqueness of Limits in a Metric Space.

Step 7 (claim 2: the shift law). Let $\pi$, $P$ and $Q$ be as in Step 6, let $t$ be real, and let $T$ be the affine datum from $3d$ to $2d$ variables with $(TZ)_{i}=Z_{i}$ and $(TZ)_{d+i}=Z_{d+i}+tZ_{2d+i}$ for $i\in[d]$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations). Then $T(X,P,Q)=(X,P+tQ)$ and $T(X_{\pi},P_{\pi},V^{1}_{\pi}\zeta)=(X_{\pi},P_{\pi}+tV^{1}_{\pi}\zeta)$ (Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §operations), so (C2) and Step 6 give $\mathrm{law}(X,P+tQ)=\mathrm{law}(X_{\pi},P_{\pi}+tV^{1}_{\pi}\zeta)=\pi\oplus t\zeta$ by The Shift of a Bounded Plan by a Self-Adjoint Field §shift. This is the shift law of claim 2.

Step 8 (claim 2: the pairing). With $\pi$, $P$ and $Q$ as in Step 6, (C2) and Step 6, with the indices $2d+j$ and $d+j$, give $\langle Q_{j},P_{j}\rangle=\langle V^{1}_{\pi}\zeta_{j},\widehat{x_{d+j}}\rangle$ for $j\in[d]$ (The Shift of a Bounded Plan by a Self-Adjoint Field §tuples); summing, $\langle Q,P\rangle_{2}=\sum_{j=1}^{d}\langle V^{1}_{\pi}\zeta_{j},\widehat{x_{d+j}}\rangle$ by Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing. The left side is real by the same clause, so it equals the real part of the right side, which is $\sum_{j}\operatorname{Re}\langle V^{1}_{\pi}\zeta_{j},\widehat{x_{d+j}}\rangle=\mathcal{J}(\zeta,\pi)$ by Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plan-pairing. This is the pairing of claim 2, which is now proved.

Step 9 (claim 3). Assume $\mathrm{law}(X,Q)=\mathrm{law}(X_{\lambda},\zeta)$. Then (C2), applied to $(X,Q)$ and $(X_{\lambda},\zeta)$ with both indices equal to $d+j$, gives $\lVert Q_{j}\rVert=\lVert\zeta_{j}\rVert$ for every $j\in[d]$, hence $\lVert Q\rVert_{2}=\lVert\zeta\rVert_{2}$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples). This proves claim 3.