Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws

In the setting of Square-Integrable Noncommutative Laws: Standing Notation, let $d,m,n\in\mathbb{N}$ 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 $J$; it is closed under sums and real multiples by Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors §self-adjoint, and it contains $\Omega$ because $J\Omega=\Omega$ by The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §conjugation. 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.

1. ($L^{2}$ tuples)¶ An $L^{2}$ $d$-tuple of $(H,M,\Omega)$ is a $d$-tuple $X=(X_{1},\dots,X_{d})$ of elements of $H_{\mathrm{sa}}$. For $L^{2}$ $d$-tuples $X$ and $Y$, $X-Y$ is the $L^{2}$ $d$-tuple $(X_{1}-Y_{1},\dots,X_{d}-Y_{d})$, and the $L^{2}$ norm $\lVert X\rVert_{2}$ of $X$ is the nonnegative square root of the nonnegative real number $\sum_{j=1}^{d}\lVert X_{j}\rVert^{2}$.

2. (Operations)¶ Let $X$ be an $L^{2}$ $d$-tuple, $Y$ an $L^{2}$ $m$-tuple and $Z$ an $L^{2}$ $n$-tuple. For an affine datum $T=(A,c)$ from $d$ to $n$ variables, $TX$ is the $L^{2}$ $n$-tuple with $(TX)_{i}=c_{i}\Omega+\sum_{j=1}^{d}A_{ij}X_{j}$ for $i\in[n]$. The pair of $X$ and $Y$ is the $L^{2}$ $(d+m)$-tuple $(X,Y)=(X_{1},\dots,X_{d},Y_{1},\dots,Y_{m})$, and the triple of $X$, $Y$ and $Z$ is the $L^{2}$ $(d+m+n)$-tuple $(X,Y,Z)=((X,Y),Z)$. For a trace-preserving embedding $\pi$ 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), $V_{\pi}X=(V_{\pi}X_{1},\dots,V_{\pi}X_{d})$, which is an $L^{2}$ $d$-tuple of $(K,N,\Psi)$ by A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry §intertwining.

3. (Law)¶ The law $\mathrm{law}(X)\in\Sigma^{2}_{d}$ of an $L^{2}$ $d$-tuple $X$ is the limit in $(\Sigma^{2}_{d},\widehat{W}_{2})$ of the sequence $(\kappa_{d}(\lambda_{s^{k}}))_{k\in\mathbb{N}}$, for any 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 $j\in[d]$. Such sequences exist, and the limit exists and does not depend on the sequence, by Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and L^2 Approximation §limit; it is unique by Uniqueness of Limits in a Metric Space. We write $\mathrm{law}(X,Y)$ for $\mathrm{law}((X,Y))$ and $\mathrm{law}(X,Y,Z)$ for $\mathrm{law}((X,Y,Z))$.