The Tracial W*-Closure of a Cyclic Tracial Operator Algebra: the Commutant of the Right Action is a Tracial W*-Probability Space Equal to the Double Commutant

In the setting of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $(H,\mathcal{A},\Omega)$ be a cyclic tracial operator algebra with trace $\tau_{\mathcal{A}}$ and conjugation $J$, and let $M=(J\mathcal{A}J)'$ be the commutant of the set $J\mathcal{A}J$ of conjugated maps. Tracial W*-probability spaces and their traces are those of Tracial W*-Probability Spaces.

1. (Containment)¶ $\mathcal{A}\subseteq M$ and $J\mathcal{A}J\subseteq M'$.

2. (Tracial W-probability space)**¶ $(H,M,\Omega)$ is a tracial W-probability space, its trace $\tau_{M}$ agrees with $\tau_{\mathcal{A}}$ on $\mathcal{A}$, and $J(T\Omega)=T^{*}\Omega$ for every $T\in M$; thus $J$ is also the conjugation of $(H,M,\Omega)$ in the sense of The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra §conjugation.

3. (Right-bounded vectors)¶ Let $\zeta\in H$ and let $C\ge0$ be real with $\lVert JSJ\zeta\rVert\le C\lVert S\Omega\rVert$ for every $S\in\mathcal{A}$. Then there is exactly one $T\in M$ with $T\Omega=\zeta$, and $\lVert T\rVert_{\mathrm{op}}\le C$.

4. (Commutant)¶ $M'=JMJ$.

5. (Double commutant)¶ $M=\mathcal{A}''$.