The Tracial Algebra of a Noncommutative Law is a Tracial W*-Probability Space: the W*-Closure of the Left Multiplications

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $d\in\mathbb{N}$ and $\lambda\in\Sigma_{d}$, and let $\mathcal{H}_{\lambda}$, $\Omega_{\lambda}$, $L_{p}$, $R_{p}$, $J_{\lambda}$, $\mathcal{M}_{\lambda}$ and $\tau_{\lambda}$ be as in Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §gns. Let $\mathcal{A}_{\lambda}=\{L_{p}:\ p\in\mathcal{P}_{d}\}$. Cyclic tracial operator algebras, tracial W*-probability spaces, their traces and conjugations, conjugated maps and commutants are those of the cited items.

1. (Left multiplications)¶ $(\mathcal{H}_{\lambda},\mathcal{A}_{\lambda},\Omega_{\lambda})$ is a cyclic tracial operator algebra, its conjugation is $J_{\lambda}$, and $J_{\lambda}\mathcal{A}_{\lambda}J_{\lambda}=\{R_{p}:\ p\in\mathcal{P}_{d}\}$.

2. (Tracial algebra)¶ $\mathcal{M}_{\lambda}=(J_{\lambda}\mathcal{A}_{\lambda}J_{\lambda})'=\mathcal{A}_{\lambda}''$, and $(\mathcal{H}_{\lambda},\mathcal{M}_{\lambda},\Omega_{\lambda})$ is a tracial W*-probability space whose trace is $\tau_{\lambda}$ and whose conjugation is $J_{\lambda}$.

3. (Law of the variables)¶ Let $L_{x}=(L_{x_{1}},\dots,L_{x_{d}})$. Then $p(L_{x})=L_{p}$ and $\langle\Omega_{\lambda},p(L_{x})\Omega_{\lambda}\rangle=\lambda(p)$ for every $p\in\mathcal{P}_{d}$.