The Tracial Algebra of a Noncommutative Law: a Norm-Closed Unital *-Algebra with a Faithful Positive Trace, Determined by Vacuum Vectors, Closed under Square Roots

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 $\lambda\in\Sigma_{d}$, 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. Write $\mathcal{H}=\mathcal{H}_{\lambda}$, $\Omega=\Omega_{\lambda}$ and $\widehat{q}$ for the complex GNS space, its vacuum vector and the classes of polynomials $q\in\mathcal{P}_{d}$; $L_{p}$, $R_{p}$ and $J=J_{\lambda}$ are the multiplication operators and the conjugation; and $\mathcal{M}=\mathcal{M}_{\lambda}$ and $\tau=\tau_{\lambda}$ are the tracial algebra and its trace.

1. (Algebra)¶ $I\in\mathcal{M}$ and $L_{p}\in\mathcal{M}$ for every $p\in\mathcal{P}_{d}$. If $S,T\in\mathcal{M}$ and $c\in\mathbb{C}$, then $S+T$, $cS$, $ST$ and $S^{*}$ belong to $\mathcal{M}$. If $(T_{k})$ is a sequence in $\mathcal{M}$ and $T_{k}\to T$ in operator norm, then $T\in\mathcal{M}$.

2. (Vacuum vectors)¶ For every $T\in\mathcal{M}$ and $q\in\mathcal{P}_{d}$: $T\widehat{q}=R_{q}T\Omega$ and $T^{*}\Omega=JT\Omega$. If $T\in\mathcal{M}$ and $T\Omega=0$, then $T=0$.

3. (Bounded vectors)¶ Let $\zeta\in\mathcal{H}$ and let $C\ge0$ be real with $\lVert R_{q}\zeta\rVert\le C\lVert\widehat{q}\rVert$ for every $q\in\mathcal{P}_{d}$. Then there is exactly one $T\in\mathcal{M}$ with $T\Omega=\zeta$, and $\lVert T\rVert_{\mathrm{op}}\le C$.

4. (Trace)¶ $\tau$ is linear, $\tau(I)=1$, $\tau(L_{p})=\lambda(p)$ for $p\in\mathcal{P}_{d}$, and for all $S,T\in\mathcal{M}$

In particular $\tau(T^{*}T)\ge0$, with equality only if $T=0$.

5. (Square roots and positivity of the trace)¶ If $T\in\mathcal{M}$ and $T\ge0$, there is $S\in\mathcal{M}$ with $S\ge0$ and $SS=T$. If $A,B\in\mathcal{M}$ with $A\ge0$ and $B\ge0$, then $\tau(AB)$ is a real number and $\tau(AB)\ge0$. For every $T\in\mathcal{M}$, $\lVert T\rVert_{\mathrm{op}}^{2}I-T^{*}T\ge0$.