Self-Adjoint Tuples in a Tracial W*-Probability Space and Their Laws

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $d\in\mathbb{N}$ 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.

1. (Self-adjoint tuples)¶ A self-adjoint $d$-tuple in $M$ is a $d$-tuple $s=(s_{1},\dots,s_{d})$ of self-adjoint elements of $M$. Its vacuum tuple is the $d$-tuple $s\Omega=(s_{1}\Omega,\dots,s_{d}\Omega)$ of elements of $H$.

2. (Law)¶ The law of a self-adjoint $d$-tuple $s$ in $M$ is the map $\lambda_{s}:\mathcal{P}_{d}\to\mathbb{C}$, $\lambda_{s}(p)=\langle\Omega,p(s)\Omega\rangle$, where $p(s)\in\mathcal{L}(H)$ is the value of $p$ at $s$ as in Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation §polynomials.