The Tracial Algebra of a Noncommutative Law and Its Trace

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation and Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $d\in\mathbb{N}$ and let $\lambda\in\Sigma_{d}$, so that $\lambda\in\Sigma_{d,r}$ for some real $r>0$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law. Let $\mathcal{H}_{\lambda}$ be the complex GNS space of $\lambda$, with vacuum vector $\Omega_{\lambda}$, and for $p\in\mathcal{P}_{d}$ let $R_{p}\in\mathcal{L}(\mathcal{H}_{\lambda})$ be the right multiplication by $p$, which is determined by $\lambda$ alone (its defining property does not involve $r$).

1. (Tracial algebra)¶ The tracial algebra of $\lambda$ is the set

2. (Trace)¶ The trace of $\lambda$ is the map $\tau_{\lambda}:\mathcal{M}_{\lambda}\to\mathbb{C}$, $\tau_{\lambda}(T)=\langle\Omega_{\lambda},T\Omega_{\lambda}\rangle_{\mathcal{H}_{\lambda}}$.