The Trace, the Conjugation and the Right Action of a Cyclic Tracial Operator Algebra

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 $\tau=\tau_{\mathcal{A}}$, let $\mathcal{A}'$ be the commutant of $\mathcal{A}$, and let conjugations and conjugated maps be those of Conjugation of a Complex Hilbert Space.

1. (Trace)¶ $\tau$ is linear and $\tau(I)=1$, and for all $S,T\in\mathcal{A}$

2. (Separation)¶ If $S\in\mathcal{A}$ and $S\Omega=0$, then $S=0$. If $T\in\mathcal{A}'$ and $T\Omega=0$, then $T=0$. In particular $\tau(S^{*}S)=0$ holds for $S\in\mathcal{A}$ only if $S=0$.

3. (Conjugation)¶ There is exactly one conjugation $J$ of $H$ with $J(S\Omega)=S^{*}\Omega$ for every $S\in\mathcal{A}$; it is called the conjugation of $(H,\mathcal{A},\Omega)$, and $J\Omega=\Omega$.

4. (Right action)¶ For every $S\in\mathcal{A}$, $JSJ\in\mathcal{A}'$ and $JSJ(T\Omega)=TS^{*}\Omega$ for every $T\in\mathcal{A}$.