Square-Integrable Tuples with a Bounded Law are Vacuum Tuples of Bounded Self-Adjoint Operators

In the setting of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation, let $(H,M,\Omega)$ be a tracial W*-probability space as in Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §spaces, let $d\in\mathbb{N}$ and let $R>0$ be real. Self-adjoint $d$-tuples $s$ in $M$, their vacuum tuples $s\Omega$ and their laws $\lambda_{s}$, $L^{2}$ $d$-tuples and their laws $\mathrm{law}(X)$ are those of Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation §tuples; $\Sigma_{d,R}$ is the set of laws with norm bound $R$ of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §laws, $\kappa_{d}$ is the canonical map of Square-Integrable Noncommutative Laws: Standing Notation §laws, and $\lVert\cdot\rVert_{\mathrm{op}}$ is the operator norm. Let $X$ be an $L^{2}$ $d$-tuple of $(H,M,\Omega)$ with $\mathrm{law}(X)=\kappa_{d}(\lambda)$ for some $\lambda\in\Sigma_{d,R}$.

1. (Operator tuple)¶ There is exactly one self-adjoint $d$-tuple $s$ in $M$ with $s\Omega=X$.

2. (Norm bound)¶ This $s$ satisfies $\lVert s_{j}\rVert_{\mathrm{op}}\le R$ for every $j\in[d]$.

3. (Law)¶ This $s$ satisfies $\lambda_{s}=\lambda$.