Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations

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 $\tau=\tau_{M}$. The letter $V$ denotes an operator, not an inner product space as in Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces.

1. (Trace-preserving embeddings)¶ A trace-preserving embedding of $(H_{0},M_{0},\Omega_{0})$ into $(H,M,\Omega)$ is a map $\pi:M_{0}\to M$ such that, for all $S,T\in M_{0}$ and $c\in\mathbb{C}$,

2. (Implementing isometry)¶ The implementing isometry of a trace-preserving embedding $\pi$ is the unique $V_{\pi}\in\mathcal{L}(H_{0},H)$ with $V_{\pi}S\Omega_{0}=\pi(S)\Omega$ for every $S\in M_{0}$, which exists by A Trace-Preserving Unital -Homomorphism between Tracial W-Probability Spaces is Implemented by an Isometry §isometry.

3. (Conditional expectation)¶ The conditional expectation of $\pi$ is the map $E_{\pi}:M\to\mathcal{L}(H_{0})$, $E_{\pi}(b)=V_{\pi}^{*}\,b\,V_{\pi}$.