Standard Form of a Tracial W*-Probability Space: the Commutation Theorem, Right-Bounded Vectors, Faithfulness and Self-Adjoint Vectors

In the setting of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $(H,M,\Omega)$ be a tracial W*-probability space with conjugation $J$, let $M'$ be the commutant of $M$, and let $JMJ$ and $JM'J$ be the sets of conjugated maps.

1. (Commutation theorem)¶ $M=(JMJ)'$, $M'=JMJ$ and $JM'J=M$.

2. (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 $S\in M$. Then there is exactly one $T\in M$ with $T\Omega=\zeta$, and $\lVert T\rVert_{\mathrm{op}}\le C$.

3. (Self-adjoint vectors)¶ Let $H_{\mathrm{sa}}=H^{J}$ be the set of fixed vectors of $J$. For all $\xi,\eta\in H_{\mathrm{sa}}$ and real $t$, the vectors $\xi+\eta$ and $t\xi$ belong to $H_{\mathrm{sa}}$ and $\langle\xi,\eta\rangle$ is real; if a sequence in $H_{\mathrm{sa}}$ converges in $H$ to $\xi$, then $\xi\in H_{\mathrm{sa}}$; $S\Omega\in H_{\mathrm{sa}}$ for every self-adjoint $S\in M$; and for every $\xi\in H_{\mathrm{sa}}$ and every real $\varepsilon>0$ there is a self-adjoint $S\in M$ with $\lVert\xi-S\Omega\rVert<\varepsilon$.