Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings

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; 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. $L^{2}$ tuples, differences $X-Y$ and the $L^{2}$ norm $\lVert\cdot\rVert_{2}$, the operations $TX$, $(X,Y)$ and $V_{\pi}X$, and laws $\mathrm{law}(X)$ are those of Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws; 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 $X$ and $Y$ be $L^{2}$ $d$-tuples of $(H,M,\Omega)$.

1. (Bounded tuples)¶ For every self-adjoint $d$-tuple $s$ in $M$, $s\Omega$ is an $L^{2}$ $d$-tuple and $\mathrm{law}(s\Omega)=\kappa_{d}(\lambda_{s})$.

2. (Lipschitz bound)¶ $\widehat{W}_{2}(\mathrm{law}(X),\mathrm{law}(Y))\le\lVert X-Y\rVert_{2}$.

3. (Moments)¶ For all $i,j\in[d]$, the numbers $\langle\Omega,X_{i}\rangle$ and $\langle X_{i},X_{j}\rangle$ are real, and

4. (Push-forwards)¶ $\mathrm{law}(TX)=T_{\#}\mathrm{law}(X)$ for every affine datum $T$ from $d$ to $n$ variables.

5. (Couplings)¶ $\mathrm{law}(X,Y)\in\Pi^{2}(\mathrm{law}(X),\mathrm{law}(Y))$ and $\mathcal{I}(\mathrm{law}(X,Y))=\lVert X-Y\rVert_{2}^{2}$.

6. (Embeddings)¶ $\mathrm{law}(V_{\pi}X)=\mathrm{law}(X)$ for every trace-preserving embedding $\pi$ of $(H,M,\Omega)$ into a tracial W*-probability space $(K,N,\Psi)$.