A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry

In the setting of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $(H_{0},M_{0},\Omega_{0})$ and $(H,M,\Omega)$ be tracial W*-probability spaces with traces $\tau_{0}=\tau_{M_{0}}$ and $\tau=\tau_{M}$ and conjugations $J_{0}$ and $J$. Let $\pi:M_{0}\to M$ be a map such that, for all $S,T\in M_{0}$ and $c\in\mathbb{C}$,

In this lemma the letter $V$ denotes an operator, not an inner product space as in Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces.

1. (Isometry)¶ There is exactly one $V\in\mathcal{L}(H_{0},H)$ with $VS\Omega_{0}=\pi(S)\Omega$ for every $S\in M_{0}$. It satisfies $V^{*}V=I_{H_{0}}$, $\lVert V\xi\rVert=\lVert\xi\rVert$ for every $\xi\in H_{0}$, and $V\Omega_{0}=\Omega$.

2. (Intertwining)¶ $\pi(S)V=VS$ and $V^{*}\pi(S)=SV^{*}$ for every $S\in M_{0}$; moreover $VJ_{0}=JV$ and $V^{*}J=J_{0}V^{*}$.

3. (Injectivity)¶ $\pi$ is injective, and $\lVert S\rVert_{\mathrm{op}}\le\lVert\pi(S)\rVert_{\mathrm{op}}$ for every $S\in M_{0}$.