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

In the setting of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation, for $\lambda\in\Sigma_{d}$ let $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ be the tracial W*-probability space of The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications §w-star and $X_{\lambda}=(\widehat{x_{1}},\dots,\widehat{x_{d}})$ the $L^{2}$ $d$-tuple of classes of the variables, as in 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. If $\lambda$ has conjugate variables $\xi_{\lambda}$, then $\xi_{\lambda}$ is an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ by Conjugate Variables, Wall Forces and Scores are Square-Integrable Tuples of the GNS Space §conjugate, and $\Phi^{*}(\lambda)$ is its free Fisher information. $\Sigma_{d,R}$ is the set of laws with norm bound $R$ of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws, and $\kappa_{d}$ the canonical map of Square-Integrable Noncommutative Laws: Standing Notation §laws; $L^{2}$ tuples, their differences, pairs, $L^{2}$ norm $\lVert\cdot\rVert_{2}$ and laws are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples, and the pairing $\langle\cdot,\cdot\rangle_{2}$ is that of Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation §pairing; convergence of real sequences is that of Limit of a Sequence of Real Numbers.

Data. Let $R>0$ and $C\ge0$ be real, let $\mu\in\Sigma_{d,R}$, and let $(\mu_{n})_{n\in\mathbb{N}}$ be a sequence in $\Sigma_{d,R}$ such that every $\mu_{n}$ has conjugate variables. Let $(H,M,\Omega)$ be a tracial W*-probability space, and let $X_{n},Q_{n}$ $(n\in\mathbb{N})$ and $X$ be $L^{2}$ $d$-tuples of $(H,M,\Omega)$ such that $\mathrm{law}(X_{n},Q_{n})=\mathrm{law}(X_{\mu_{n}},\xi_{\mu_{n}})$ and $\lVert Q_{n}\rVert_{2}\le C$ for every $n$, $\mathrm{law}(X)=\kappa_{d}(\mu)$, and $\lVert X_{n}-X\rVert_{2}\to0$.

1. (Conjugate variables of the limit)¶ $\mu$ has conjugate variables, and $\Phi^{*}(\mu)\le C^{2}$.

2. (Projected convergence)¶ Let $\xi_{\mu}$ be the conjugate variables of $\mu$ given by claim 1, an $L^{2}$ $d$-tuple of $(\mathcal{H}_{\mu},\mathcal{M}_{\mu},\Omega_{\mu})$ by Conjugate Variables, Wall Forces and Scores are Square-Integrable Tuples of the GNS Space §conjugate. For every $L^{2}$ $d$-tuple $\eta$ of $(\mathcal{H}_{\mu},\mathcal{M}_{\mu},\Omega_{\mu})$ and every $L^{2}$ $d$-tuple $Y$ of $(H,M,\Omega)$ with $\mathrm{law}(X,Y)=\mathrm{law}(X_{\mu},\eta)$,

$$\langle Q_{n},Y\rangle_{2}\to\langle\xi_{\mu},\eta\rangle_{2}.$$