Proof of Projected Closedness of Conjugate Variables along Square-Integrable Realisations with Bounded Free Fisher Information

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

Conventions. (C1) Inner products and pairings. Inner products are linear in the second argument and conjugate-linear in the first (Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces); accordingly, in The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §conjugate the map $p\mapsto\langle\xi_{j},\widehat{p}\rangle$ is the linear map $\partial^{\lambda}_{j}$. Let $(K,N,\Psi)$ be a tracial W*-probability space and $K^{d}$ the complex Hilbert space of $d$-tuples of Finite Direct Sums of a Complex Hilbert Space: the Hilbert Structure, Coordinate Inclusions, Block Entries of Bounded Operators and Commutation with Diagonal Operators §hilbert. By Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space §pairing, for $L^{2}$ $d$-tuples $Z,W$ of $(K,N,\Psi)$ the pairing $\langle Z,W\rangle_{2}=\sum_{j=1}^{d}\langle Z_{j},W_{j}\rangle$ is the inner product of $Z$ and $W$ in $K^{d}$ and is a real number, so that $\langle Z,W\rangle_{2}=\sum_{j=1}^{d}\operatorname{Re}\langle Z_{j},W_{j}\rangle$; and $\lVert Z\rVert_{2}$ is the norm of $Z$ in $K^{d}$, while the difference $Z-W$ of Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples is the difference in $K^{d}$. Hence: by the Cauchy--Schwarz inequality in $K^{d}$,

$$|\langle Z,W\rangle_{2}|\le\lVert Z\rVert_{2}\,\lVert W\rVert_{2};\tag{CS}$$

the pairing is additive in its second argument, $\langle Z,W-W'\rangle_{2}=\langle Z,W\rangle_{2}-\langle Z,W'\rangle_{2}$, by linearity of the inner product of $K^{d}$; expanding $\lVert Z_{j}-W_{j}\rVert^{2}=\lVert Z_{j}\rVert^{2}-2\operatorname{Re}\langle Z_{j},W_{j}\rangle+\lVert W_{j}\rVert^{2}$ and summing over $j$ (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples),

$$\lVert Z-W\rVert_{2}^{2}=\lVert Z\rVert_{2}^{2}-2\langle Z,W\rangle_{2}+\lVert W\rVert_{2}^{2};\tag{EX}$$

and $\lVert Z\rVert_{2}\le\sum_{j=1}^{d}\lVert Z_{j}\rVert$, since the square of the right side is at least $\sum_{j}\lVert Z_{j}\rVert^{2}=\lVert Z\rVert_{2}^{2}$.

(C2) Evaluation. For a tuple $T$ of bounded operators and $p\in\mathcal{P}_{m}$, the value $p(T)$ is that of Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §evaluation: it is linear in $p$ with $x_{w}(T)=T_{w}$. Since $p=\sum_{w\in\operatorname{supp}p}p(w)\,x_{w}$ (both sides are maps $W_{m}\to\mathbb{C}$ with the same value at every word, by The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §linear and The Algebra of Noncommutative Polynomials in Finitely Many Self-Adjoint Variables §monomials), we get $p(T)=\sum_{w\in\operatorname{supp}p}p(w)\,T_{w}$. Moreover $(pq)(T)=p(T)q(T)$, $x_{j}(T)=T_{j}$ and $(\sigma_{a}(p))(S)=p(a(S))$ by 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 §values and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §substitution, and $p^{*}(T)=p(T)^{*}$ when the entries of $T$ are self-adjoint, by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §adjoint. If $T$ is a self-adjoint tuple in $N$, then $p(T)\in N$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §transport, applied with $\mathcal{A}=N$ and $\Phi$ the identity map of $N$: indeed $N=(N')'$ (Tracial W*-Probability Spaces §space) is a commutant, so it contains $I$ and is closed under sums, complex multiples and composition by Basic Properties of Commutants: Unital Algebras Closed under Weak Limits, Order Reversal, the Triple Commutant, and Conjugation §algebra; and $p(T)$ is self-adjoint if moreover $p$ is self-adjoint. For a self-adjoint $m$-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 $p\in\mathcal{P}_{m}$

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

Step 1 (the laws of the positions). Let $\lambda\in\Sigma_{d,r}$ for some real $r>0$, and let $\ell=(L_{x_{1}},\dots,L_{x_{d}})$ be the tuple of left multiplications on $\mathcal{H}_{\lambda}$. Each $L_{x_{i}}$ lies in $\mathcal{A}_{\lambda}\subseteq\mathcal{A}_{\lambda}''=\mathcal{M}_{\lambda}$ by Basic Properties of Commutants: Unital Algebras Closed under Weak Limits, Order Reversal, the Triple Commutant, and Conjugation §order and The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star, and it 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, as $x_{i}^{*}=x_{i}$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint). So $\ell$ is a self-adjoint $d$-tuple in $\mathcal{M}_{\lambda}$, with $\ell\Omega_{\lambda}=(\widehat{x_{1}},\dots,\widehat{x_{d}})=X_{\lambda}$ 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 §vacuum, and $\lambda_{\ell}(p)=\langle\Omega_{\lambda},p(\ell)\Omega_{\lambda}\rangle=\lambda(p)$ for every $p$ by Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws §law and The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §law. Since $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ is a tracial W*-probability space (The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star), Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded gives

$$\mathrm{law}(X_{\lambda})=\kappa_{d}(\lambda).\tag{1.1}$$

Now fix $n$. The marginal datum $\mathrm{pr}^{1}$ (Affine Data and Affine Substitutions of Noncommutative Polynomials §coordinate) satisfies $\mathrm{pr}^{1}(Z,W)=Z$ for $L^{2}$ $d$-tuples $Z,W$ of any tracial W*-probability space, by Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §operations (its constant part is $0$ and $P^{1}_{ij}=1$ exactly for $j=i$). By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §push-forward, applied in $(H,M,\Omega)$ and in $(\mathcal{H}_{\mu_{n}},\mathcal{M}_{\mu_{n}},\Omega_{\mu_{n}})$ to the $2d$-tuples $(X_{n},Q_{n})$ and $(X_{\mu_{n}},\xi_{\mu_{n}})$ (the latter is an $L^{2}$ $2d$-tuple because $X_{\mu_{n}}$, the vacuum tuple of the self-adjoint tuple $\ell$ above for $\lambda=\mu_{n}$, is an $L^{2}$ $d$-tuple by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law, and $\xi_{\mu_{n}}$ is an $L^{2}$ $d$-tuple by Conjugate Variables, Wall Forces and Scores are Square-Integrable Tuples of the GNS Space §conjugate), and by (1.1) with $\lambda=\mu_{n}\in\Sigma_{d,R}$,

$$\mathrm{law}(X_{n})=\mathrm{pr}^{1}_{\#}\mathrm{law}(X_{n},Q_{n})=\mathrm{pr}^{1}_{\#}\mathrm{law}(X_{\mu_{n}},\xi_{\mu_{n}})=\mathrm{law}(X_{\mu_{n}})=\kappa_{d}(\mu_{n}).\tag{1.2}$$

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 are self-adjoint $d$-tuples $s^{n}$ and $s$ in $M$ with $s^{n}\Omega=X_{n}$, $s\Omega=X$, $\lVert s^{n}_{j}\rVert_{\mathrm{op}}\le R$, $\lVert s_{j}\rVert_{\mathrm{op}}\le R$ for all $j\in[d]$, $\lambda_{s^{n}}=\mu_{n}$ and $\lambda_{s}=\mu$.

Step 2 (Wasserstein convergence and claim 1). By Square-Integrable Noncommutative Laws: the Wasserstein Completion of the Laws, Affine Push-Forwards, Moments, Couplings and Cost §laws, $(\Sigma^{2}_{d},\widehat{W}_{2})$ is the metric completion of $(\Sigma_{d},W_{2})$ with canonical map $\kappa_{d}$, so The Metric Completion is a Complete Metric Space with a Dense Isometric Copy of the Space, and Maps Preserving Cauchy Sequences Extend to It §isometry, (1.2), the hypothesis $\mathrm{law}(X)=\kappa_{d}(\mu)$ and Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §lipschitz give

$$W_{2}(\mu_{n},\mu)=\widehat{W}_{2}(\kappa_{d}(\mu_{n}),\kappa_{d}(\mu))=\widehat{W}_{2}(\mathrm{law}(X_{n}),\mathrm{law}(X))\le\lVert X_{n}-X\rVert_{2}\qquad(n\in\mathbb{N}).\tag{2.1}$$

As $\lVert X_{n}-X\rVert_{2}\to0$ and $W_{2}(\mu_{n},\mu)\ge0$, the real sequence $(W_{2}(\mu_{n},\mu))_{n}$ converges to $0$ (Limit of a Sequence of Real Numbers). All $\mu_{n}$ and $\mu$ lie in $\Sigma_{d,R}$, so $\mu_{n}\to\mu$ weak-star by Moments of Noncommutative Laws are Lipschitz in the Wasserstein Distance §convergence. Next, for each $n$, 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 §norm applies with $r=R$, $\lambda=\mu_{n}$, $\zeta=\xi_{\mu_{n}}$ (an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\mu_{n}},\mathcal{M}_{\mu_{n}},\Omega_{\mu_{n}})$ by Conjugate Variables, Wall Forces and Scores are Square-Integrable Tuples of the GNS Space §conjugate), the space $(H,M,\Omega)$, $X=X_{n}$ and $Q=Q'=Q_{n}$: indeed $\mathrm{law}(X_{n})=\kappa_{d}(\mu_{n})$ by (1.2) and $\mathrm{law}(X_{n},Q_{n})=\mathrm{law}(X_{\mu_{n}},\xi_{\mu_{n}})$. Hence $\lVert\xi_{\mu_{n}}\rVert_{2}=\lVert Q_{n}\rVert_{2}\le C$, and by The Free Fisher Information of a Noncommutative Law §fisher and Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples

$$\Phi^{*}(\mu_{n})=\sum_{j=1}^{d}\lVert\xi_{\mu_{n},j}\rVert^{2}=\lVert\xi_{\mu_{n}}\rVert_{2}^{2}\le C^{2}.$$

Now Free Fisher Information: the Cramer-Rao Inequality, the Semicircular Laws, and Closedness under Weak-Star Limits §closed, with $c=C^{2}$ and the sequence $(\mu_{n})_{n}$ in $\Sigma_{d}$ converging weak-star to $\mu\in\Sigma_{d}$, shows that $\mu$ has conjugate variables $\xi_{\mu}$ and $\Phi^{*}(\mu)\le C^{2}$. This is claim 1. Note also $\lVert\xi_{\mu}\rVert_{2}^{2}=\Phi^{*}(\mu)\le C^{2}$, so $\lVert\xi_{\mu}\rVert_{2}\le C$.

Step 3 (a Lipschitz bound for polynomial evaluation). Let $b\in M$ and let $c\in M$ be self-adjoint. By 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, $JcJ(b\Omega)-cb\Omega=(bc-cb)\Omega=bc\Omega-cb\Omega$, where $J$ is the conjugation of $(H,M,\Omega)$; hence $bc\Omega=JcJ(b\Omega)$, and since $\lVert JcJ\rVert_{\mathrm{op}}=\lVert c\rVert_{\mathrm{op}}$ by Basic Properties of Commutants: Unital Algebras Closed under Weak Limits, Order Reversal, the Triple Commutant, and Conjugation §conjugation ($J$ is a conjugation of $H$ by The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §conjugation),

$$\lVert bc\Omega\rVert\le\lVert c\rVert_{\mathrm{op}}\,\lVert b\Omega\rVert.$$

By induction on $m$, applying this to $bc_{1}\cdots c_{m-1}\in M$ and $c_{m}$ (products of elements of $M$ lie in $M$ by Basic Properties of Commutants: Unital Algebras Closed under Weak Limits, Order Reversal, the Triple Commutant, and Conjugation §algebra), $\lVert bc_{1}\cdots c_{m}\Omega\rVert\le R^{m}\lVert b\Omega\rVert$ for every $b\in M$ and all self-adjoint $c_{1},\dots,c_{m}\in M$ of operator norm at most $R$. Now let $a,a'$ be self-adjoint $d$-tuples in $M$ whose entries have operator norm at most $R$, and let $w=(i_{1},\dots,i_{k})$ be a word of length $k\ge1$. Then $a_{w}-a'_{w}$ is the sum over $u\in[k]$ of $a_{i_{1}}\cdots a_{i_{u-1}}(a_{i_{u}}-a'_{i_{u}})a'_{i_{u+1}}\cdots a'_{i_{k}}$ (a telescoping sum; the left factor is $I$ for $u=1$ and the right factor is $I$ for $u=k$). The left factor is the product along the word $(i_{1},\dots,i_{u-1})$ and 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 (it is $I$ for $u=1$), and the bound just proved with $b=a_{i_{u}}-a'_{i_{u}}\in M$ bounds the norm of the rest applied to $\Omega$ by $R^{k-u}\lVert a_{i_{u}}\Omega-a'_{i_{u}}\Omega\rVert\le R^{k-u}\lVert a\Omega-a'\Omega\rVert_{2}$. Hence $\lVert a_{w}\Omega-a'_{w}\Omega\rVert\le kR^{k-1}\lVert a\Omega-a'\Omega\rVert_{2}$, and $a_{\varnothing}-a'_{\varnothing}=I-I=0$. By (C2), for every $p\in\mathcal{P}_{d}$,

$$\lVert p(a)\Omega-p(a')\Omega\rVert\le K_{p}\,\lVert a\Omega-a'\Omega\rVert_{2},\qquad K_{p}=\sum_{w\in\operatorname{supp}p,\ w\neq\varnothing}|p(w)|\,|w|\,R^{|w|-1},\tag{3.1}$$

where $|w|$ is the length of $w$.

Step 4 (pairings with polynomial fields of the positions). Let $\lambda\in\Sigma_{d,R}$, let $\zeta$ be an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$, let $Z$ and $W$ be $L^{2}$ $d$-tuples of $(H,M,\Omega)$ with $\mathrm{law}(Z)=\kappa_{d}(\lambda)$ and $\mathrm{law}(Z,W)=\mathrm{law}(X_{\lambda},\zeta)$, let $z$ be a self-adjoint $d$-tuple in $M$ with $z\Omega=Z$ and $\lambda_{z}=\lambda$, and let $q=(q_{1},\dots,q_{d})$ be a $d$-tuple of self-adjoint polynomials in $\mathcal{P}_{d}$. We show that $q(z)\Omega=(q_{1}(z)\Omega,\dots,q_{d}(z)\Omega)$ and $\widehat{q}=(\widehat{q_{1}},\dots,\widehat{q_{d}})$ are $L^{2}$ $d$-tuples of $(H,M,\Omega)$ and of $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ respectively, and

$$\langle W,q(z)\Omega\rangle_{2}=\langle\zeta,\widehat{q}\rangle_{2},\qquad\lVert W-q(z)\Omega\rVert_{2}=\lVert\zeta-\widehat{q}\rVert_{2}.\tag{4.1}$$

(a) The graph plan. By (C2), $t=(z_{1},\dots,z_{d},q_{1}(z),\dots,q_{d}(z))$ is a self-adjoint $2d$-tuple in $M$, with vacuum tuple $t\Omega=(Z,q(z)\Omega)$; the entries $q_{j}(z)\Omega$ are fixed by $J$ by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §law, so $q(z)\Omega$ is an $L^{2}$ $d$-tuple (Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws §tuples). Put $\pi=\lambda_{t}$, 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 Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §coupling, $\pi\in\Pi(\lambda_{z},\lambda_{q(z)})$, so $\pi\circ\iota^{1}=\lambda_{z}=\lambda$ by Couplings of Two Noncommutative Laws and Their Quadratic Cost §coupling; thus $\pi$ is a bounded plan at $\lambda$. By Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings §bounded, $\mathrm{law}(Z,q(z)\Omega)=\mathrm{law}(t\Omega)=\kappa_{2d}(\pi)$.

(b) The field side. Each $q_{j}$ is self-adjoint, so $J_{\lambda}\widehat{q_{j}}=\widehat{q_{j}^{*}}=\widehat{q_{j}}$ 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 §conjugation; as $J_{\lambda}$ is the conjugation of $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ (The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star), $\widehat{q}$ is an $L^{2}$ $d$-tuple. Let $j\in[d]$ and $g=x_{d+j}-\iota^{1}(q_{j})\in\mathcal{P}_{2d}$. As $\iota^{1}=\sigma_{(x_{1},\dots,x_{d})}$ (Couplings of Two Noncommutative Laws and Their Quadratic Cost §marginals), (C2) gives $(\iota^{1}(q_{j}))(t)=q_{j}(t_{1},\dots,t_{d})=q_{j}(z)$ and $x_{d+j}(t)=t_{d+j}=q_{j}(z)$, so $g(t)=0$. The law $\pi$ has some norm bound (Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law), so by the linearity of the class map (The Complex Hilbert Completion is a Complex Hilbert Space Containing a Dense Isometric Image, and Bounded Complex-Linear Maps Extend to It §isometry, The Complex GNS Space of a Tracial State on Noncommutative Polynomials §classes), 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 (E),

$$\lVert\widehat{x_{d+j}}-\widehat{\iota^{1}(q_{j})}\rVert^{2}=\lVert\widehat{g}\rVert^{2}=\pi(g^{*}g)=\lambda_{t}(g^{*}g)=\lVert g(t)\Omega\rVert^{2}=0$$

in $\mathcal{H}_{\pi}$. With the marginal isometry $V^{1}_{\pi}$, which satisfies $V^{1}_{\pi}\widehat{q_{j}}=\widehat{\iota^{1}(q_{j})}$ by that clause and $(V^{1}_{\pi})^{*}V^{1}_{\pi}=I$ by Marginals of a Noncommutative Law: the Isometry of GNS Spaces, the Trace-Preserving Embedding of Tracial Algebras and the Conditional Expectation §isometry, we get $\widehat{x_{d+j}}=V^{1}_{\pi}\widehat{q_{j}}$ and $\langle V^{1}_{\pi}\zeta_{j},\widehat{x_{d+j}}\rangle=\langle\zeta_{j},(V^{1}_{\pi})^{*}V^{1}_{\pi}\widehat{q_{j}}\rangle=\langle\zeta_{j},\widehat{q_{j}}\rangle$.

(c) The pairing. The data 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 are met with $r=R$, $\lambda$, $\zeta$, the space $(H,M,\Omega)$, $X=Z$ and $Q=Q'=W$, and claim 2 there applies with the bounded plan $\pi$ and $P=q(z)\Omega$, since $\mathrm{law}(Z,W)=\mathrm{law}(X_{\lambda},\zeta)$ and $\mathrm{law}(Z,P)=\kappa_{2d}(\pi)$ by (a). By 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 §pairing, Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws §plan-pairing, (b) and (C1),

$$\langle W,q(z)\Omega\rangle_{2}=\mathcal{J}(\zeta,\pi)=\sum_{j=1}^{d}\operatorname{Re}\langle V^{1}_{\pi}\zeta_{j},\widehat{x_{d+j}}\rangle=\sum_{j=1}^{d}\operatorname{Re}\langle\zeta_{j},\widehat{q_{j}}\rangle=\langle\zeta,\widehat{q}\rangle_{2}.$$

(d) The distance. By 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 §norm (same data), $\lVert W\rVert_{2}=\lVert\zeta\rVert_{2}$. By (E), $\lambda_{z}=\lambda$ 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 §vacuum, $\lVert q_{j}(z)\Omega\rVert^{2}=\lambda(q_{j}^{*}q_{j})=\lVert\widehat{q_{j}}\rVert^{2}$ for every $j$, so $\lVert q(z)\Omega\rVert_{2}=\lVert\widehat{q}\rVert_{2}$. Applying (EX) in $(H,M,\Omega)$ and in $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ and using (c),

$$\lVert W-q(z)\Omega\rVert_{2}^{2}=\lVert W\rVert_{2}^{2}-2\langle W,q(z)\Omega\rangle_{2}+\lVert q(z)\Omega\rVert_{2}^{2}=\lVert\zeta\rVert_{2}^{2}-2\langle\zeta,\widehat{q}\rangle_{2}+\lVert\widehat{q}\rVert_{2}^{2}=\lVert\zeta-\widehat{q}\rVert_{2}^{2},$$

and both norms are nonnegative, which proves (4.1).

Step 5 (convergence of the difference quotients). Let $q=(q_{1},\dots,q_{d})$ be a $d$-tuple of self-adjoint polynomials in $\mathcal{P}_{d}$. For a word $u$ put $\rho_{u}=1$ if $u=\varnothing$ and $\rho_{u}=R^{|u|}$ otherwise; every $\lambda\in\Sigma_{d,R}$ satisfies $|\lambda(x_{u})|\le\rho_{u}$, by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §norm-bound for $u\neq\varnothing$ and by condition (a) of Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state for $u=\varnothing$ ($x_{\varnothing}=1$). By Moments of Noncommutative Laws are Lipschitz in the Wasserstein Distance §lipschitz, applied to $p=x_{u}$, choose for every word $u$ a real $c_{u}\ge0$ with $|\nu(x_{u})-\nu'(x_{u})|\le c_{u}W_{2}(\nu,\nu')$ for all $\nu,\nu'\in\Sigma_{d,R}$. For words $u,v$ and $n\in\mathbb{N}$,

$$|\mu_{n}(x_{u})\mu_{n}(x_{v})-\mu(x_{u})\mu(x_{v})|\le|\mu_{n}(x_{u})|\,|\mu_{n}(x_{v})-\mu(x_{v})|+|\mu(x_{v})|\,|\mu_{n}(x_{u})-\mu(x_{u})|\le(\rho_{u}c_{v}+\rho_{v}c_{u})\,W_{2}(\mu_{n},\mu).$$

By linearity of $\partial^{\lambda}_{j}$ and The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §difference-quotient, $\partial^{\lambda}_{j}(q_{j})=\sum_{w\in\operatorname{supp}q_{j},\,w\neq\varnothing}q_{j}(w)\sum_{l=1}^{|w|}\delta_{w_{l}j}\lambda(x_{w_{<l}})\lambda(x_{w_{>l}})$ for $\lambda\in\Sigma_{d}$. Hence, with the real constant

$$K'_{q}=\sum_{j=1}^{d}\ \sum_{w\in\operatorname{supp}q_{j},\,w\neq\varnothing}|q_{j}(w)|\sum_{l=1}^{|w|}\bigl(\rho_{w_{<l}}c_{w_{>l}}+\rho_{w_{>l}}c_{w_{<l}}\bigr)\ge0,$$

which depends only on $q$ and $R$, and using $|\operatorname{Re}\,y|\le|y|$ for complex $y$,

$$\Bigl|\sum_{j=1}^{d}\operatorname{Re}\partial^{\mu_{n}}_{j}(q_{j})-\sum_{j=1}^{d}\operatorname{Re}\partial^{\mu}_{j}(q_{j})\Bigr|\le\sum_{j=1}^{d}\bigl|\partial^{\mu_{n}}_{j}(q_{j})-\partial^{\mu}_{j}(q_{j})\bigr|\le K'_{q}\,W_{2}(\mu_{n},\mu)\le K'_{q}\lVert X_{n}-X\rVert_{2},\tag{5.1}$$

the last inequality by (2.1).

Step 6 (the two sides at a polynomial field). Let $q$ be as in Step 5, and write $P=q(s)\Omega$ and $P_{n}=q(s^{n})\Omega$, $L^{2}$ $d$-tuples of $(H,M,\Omega)$ by Step 4(a), which uses only that $s$ and $s^{n}$ are self-adjoint tuples in $M$. Step 4 with $\lambda=\mu_{n}$, $\zeta=\xi_{\mu_{n}}$, $Z=X_{n}$, $W=Q_{n}$ and $z=s^{n}$ (its hypotheses hold by (1.2), the hypothesis on $\mathrm{law}(X_{n},Q_{n})$, Step 1 and Conjugate Variables, Wall Forces and Scores are Square-Integrable Tuples of the GNS Space §conjugate), then (C1) and The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §conjugate for $\mu_{n}$, give

$$\langle Q_{n},P_{n}\rangle_{2}=\langle\xi_{\mu_{n}},\widehat{q}\rangle_{2}=\sum_{j=1}^{d}\operatorname{Re}\langle\xi_{\mu_{n},j},\widehat{q_{j}}\rangle=\sum_{j=1}^{d}\operatorname{Re}\partial^{\mu_{n}}_{j}(q_{j}),$$

and likewise, by (C1) and The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §conjugate for $\mu$ (claim 1), $\langle\xi_{\mu},\widehat{q}\rangle_{2}=\sum_{j=1}^{d}\operatorname{Re}\partial^{\mu}_{j}(q_{j})$. By (5.1),

$$|\langle Q_{n},P_{n}\rangle_{2}-\langle\xi_{\mu},\widehat{q}\rangle_{2}|\le K'_{q}\lVert X_{n}-X\rVert_{2}.\tag{6.1}$$

By (C1), (3.1) with $a=s$, $a'=s^{n}$ (entries of operator norm at most $R$ by Step 1), and $s\Omega-s^{n}\Omega=X-X_{n}$,

$$\lVert P-P_{n}\rVert_{2}\le\sum_{j=1}^{d}\lVert q_{j}(s)\Omega-q_{j}(s^{n})\Omega\rVert\le K_{q}\lVert X-X_{n}\rVert_{2},\qquad K_{q}=\sum_{j=1}^{d}K_{q_{j}}.\tag{6.2}$$

Step 7 (claim 2). Let $\eta$ and $Y$ be as in claim 2, and let $\varepsilon>0$ be real. The choices are made in this order: first $\varepsilon$, then the polynomials $q_{j}$ (depending on $\varepsilon$ and $\eta$), then $N$ (depending on $\varepsilon$ and $q$).

Choice of $q$. Let $j\in[d]$. The classes are dense in $\mathcal{H}_{\mu}$ 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, so there is $p_{j}\in\mathcal{P}_{d}$ with $\lVert\widehat{p_{j}}-\eta_{j}\rVert<\varepsilon/d$. Put $q_{j}=\tfrac12(p_{j}+p_{j}^{*})$, a self-adjoint polynomial by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §self-adjoint. 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), $J_{\mu}\widehat{p_{j}}=\widehat{p_{j}^{*}}$ and $J_{\mu}$ is additive 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), and $J_{\mu}\eta_{j}=\eta_{j}$ because $\eta$ is an $L^{2}$ $d$-tuple and $J_{\mu}$ is the conjugation of $(\mathcal{H}_{\mu},\mathcal{M}_{\mu},\Omega_{\mu})$ (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). Hence

$$\widehat{q_{j}}-\eta_{j}=\tfrac12\bigl((\widehat{p_{j}}-\eta_{j})+J_{\mu}(\widehat{p_{j}}-\eta_{j})\bigr),\qquad\lVert\widehat{q_{j}}-\eta_{j}\rVert\le\lVert\widehat{p_{j}}-\eta_{j}\rVert<\varepsilon/d,$$

and by (C1) $\lVert\widehat{q}-\eta\rVert_{2}\le\sum_{j}\lVert\widehat{q_{j}}-\eta_{j}\rVert<\varepsilon$. Step 4 with $\lambda=\mu$, $\zeta=\eta$, $Z=X$, $W=Y$ and $z=s$ (its hypotheses hold by the hypotheses $\mathrm{law}(X)=\kappa_{d}(\mu)$, $\mathrm{law}(X,Y)=\mathrm{law}(X_{\mu},\eta)$, and Step 1) gives, with $P=q(s)\Omega$,

$$\lVert Y-P\rVert_{2}=\lVert\eta-\widehat{q}\rVert_{2}<\varepsilon.\tag{7.1}$$

Choice of $N$. As $\lVert X_{n}-X\rVert_{2}\to0$, choose $N$ such that $\lVert X_{n}-X\rVert_{2}<\varepsilon/(CK_{q}+K'_{q}+1)$ for all $n\ge N$.

Estimate. Let $n\ge N$. By the additivity of the pairing in its second argument (C1),

$$\langle Q_{n},Y\rangle_{2}-\langle\xi_{\mu},\eta\rangle_{2}=\langle Q_{n},Y-P\rangle_{2}+\langle Q_{n},P-P_{n}\rangle_{2}+\bigl(\langle Q_{n},P_{n}\rangle_{2}-\langle\xi_{\mu},\widehat{q}\rangle_{2}\bigr)+\langle\xi_{\mu},\widehat{q}-\eta\rangle_{2}.$$

By (CS) with $\lVert Q_{n}\rVert_{2}\le C$ and (7.1), the first term has modulus at most $C\varepsilon$; by (CS) and (6.2), the second at most $CK_{q}\lVert X_{n}-X\rVert_{2}$; by (6.1), the third at most $K'_{q}\lVert X_{n}-X\rVert_{2}$; and by (CS) in $(\mathcal{H}_{\mu},\mathcal{M}_{\mu},\Omega_{\mu})$, $\lVert\xi_{\mu}\rVert_{2}\le C$ (Step 2) and (7.1), the fourth at most $C\varepsilon$. Hence

$$|\langle Q_{n},Y\rangle_{2}-\langle\xi_{\mu},\eta\rangle_{2}|\le2C\varepsilon+(CK_{q}+K'_{q})\lVert X_{n}-X\rVert_{2}<(2C+1)\varepsilon\qquad(n\ge N).$$

Given any real $\varepsilon'>0$, running this argument with $\varepsilon=\varepsilon'/(2C+1)$ yields $N$ with $|\langle Q_{n},Y\rangle_{2}-\langle\xi_{\mu},\eta\rangle_{2}|<\varepsilon'$ for all $n\ge N$. By Limit of a Sequence of Real Numbers, $\langle Q_{n},Y\rangle_{2}\to\langle\xi_{\mu},\eta\rangle_{2}$, which is claim 2.