Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and $L^2$ Approximation

In the setting of Square-Integrable Noncommutative Laws: Standing Notation, let $d,n\in\mathbb{N}$ and let $(H,M,\Omega)$ be a tracial W*-probability space with conjugation $J$; the letter $M$ names this set of operators, while $M(\lambda)$ remains the second moment of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws. Let $H_{\mathrm{sa}}=H^{J}$ be the set of fixed vectors of $J$. Self-adjoint tuples $s$ in $M$, their vacuum tuples $s\Omega$ and their laws $\lambda_{s}$ are those of Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws. Let $s$ and $t$ be self-adjoint $d$-tuples in $M$.

1. (Law)¶ $\lambda_{s}\in\Sigma_{d,R}$ for every real $R>0$ with $\lVert s_{j}\rVert_{\mathrm{op}}\le R$ for every $j\in[d]$; in particular $\lambda_{s}\in\Sigma_{d}$. Moreover $s_{j}\Omega\in H_{\mathrm{sa}}$ for every $j\in[d]$.

2. (Moments)¶ For all $i,j\in[d]$, $\lambda_{s}(x_{i})=\langle\Omega,s_{i}\Omega\rangle$ and $\lambda_{s}(x_{i}x_{j})=\langle s_{i}\Omega,s_{j}\Omega\rangle$, and both numbers are real.

3. (Affine images)¶ Let $T=(A,c)$ be an affine datum from $d$ to $n$ variables, and let $u$ be the $n$-tuple with $u_{i}=c_{i}I+\sum_{j=1}^{d}A_{ij}s_{j}$ for $i\in[n]$. Then $u$ is a self-adjoint $n$-tuple in $M$, $u_{i}\Omega=c_{i}\Omega+\sum_{j=1}^{d}A_{ij}s_{j}\Omega$ for every $i\in[n]$, and $\lambda_{u}=\lambda_{s}\circ\sigma_{T}$.

4. (Couplings)¶ The $2d$-tuple $(s,t)=(s_{1},\dots,s_{d},t_{1},\dots,t_{d})$ is a self-adjoint $2d$-tuple in $M$, $\lambda_{(s,t)}\in\Pi(\lambda_{s},\lambda_{t})$, and

5. (Embeddings)¶ Let $\pi$ be a trace-preserving embedding of $(H,M,\Omega)$ into a tracial W*-probability space $(K,N,\Psi)$, with implementing isometry $V_{\pi}$, the letter $V$ denoting an operator, not an inner product space as in Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces. Then $\pi(s)=(\pi(s_{1}),\dots,\pi(s_{d}))$ is a self-adjoint $d$-tuple in $N$, $\pi(s_{j})\Psi=V_{\pi}s_{j}\Omega$ for every $j\in[d]$, and $\lambda_{\pi(s)}=\lambda_{s}$.

6. (Approximation)¶ Let $X=(X_{1},\dots,X_{d})$ be a $d$-tuple of elements of $H_{\mathrm{sa}}$. There is a sequence $(s^{k})_{k\in\mathbb{N}}$ of self-adjoint $d$-tuples in $M$ such that $(s^{k}_{j}\Omega)_{k\in\mathbb{N}}$ converges to $X_{j}$ in $H$ for every $j\in[d]$. For every such sequence, $(\kappa_{d}(\lambda_{s^{k}}))_{k\in\mathbb{N}}$ converges in the metric space $(\Sigma^{2}_{d},\widehat{W}_{2})$, and its limit is the same for all such sequences.