The Conditional Expectation of a Trace-Preserving Embedding: Values in the Subalgebra, Bimodule Property, Trace, Positivity, Contraction and the Jones Projection

In the setting of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $\pi$ be a trace-preserving embedding of a tracial W*-probability space $(H_{0},M_{0},\Omega_{0})$ into a tracial W*-probability space $(H,M,\Omega)$, with traces $\tau_{0}=\tau_{M_{0}}$ and $\tau=\tau_{M}$, implementing isometry $V=V_{\pi}$ and conditional expectation $E=E_{\pi}$. The letter $V$ denotes an operator, not an inner product space as in Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces.

1. (Values)¶ $E(b)\in M_{0}$ for every $b\in M$; $E$ is linear; and $E(b^{*})=E(b)^{*}$ and $\lVert E(b)\rVert_{\mathrm{op}}\le\lVert b\rVert_{\mathrm{op}}$ for every $b\in M$.

2. (Bimodule property)¶ $E(\pi(S)\,b\,\pi(T))=S\,E(b)\,T$ for all $b\in M$ and $S,T\in M_{0}$; in particular $E(\pi(S))=S$ for every $S\in M_{0}$, and $E(I)=I$.

3. (Trace)¶ $\tau_{0}(E(b))=\tau(b)$ and $\tau_{0}(S\,E(b))=\tau(\pi(S)\,b)$ for all $b\in M$ and $S\in M_{0}$.

4. (Positivity)¶ $E(b^{*}b)\ge0$ for every $b\in M$.

5. (Jones projection)¶ The operator $e=VV^{*}\in\mathcal{L}(H)$ is an orthogonal projection, and $e\,\pi(S)\Omega=\pi(S)\Omega$ for every $S\in M_{0}$. For every $b\in M$, $E(b)\Omega_{0}=V^{*}b\Omega$ and $e\,b\Omega=\pi(E(b))\Omega$.