Trace-Preserving Embeddings Preserve Operator Norms and Compose

In the setting of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $(H_{0},M_{0},\Omega_{0})$, $(H_{1},M_{1},\Omega_{1})$ and $(H_{2},M_{2},\Omega_{2})$ be tracial W*-probability spaces, let $\pi$ be a trace-preserving embedding of $(H_{0},M_{0},\Omega_{0})$ into $(H_{1},M_{1},\Omega_{1})$, and let $\pi'$ be a trace-preserving embedding of $(H_{1},M_{1},\Omega_{1})$ into $(H_{2},M_{2},\Omega_{2})$, with implementing isometries $V_{\pi}$ and $V_{\pi'}$. The letter $V$ denotes an operator, not an inner product space as in Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces.

1. (Operator norm)¶ $\lVert\pi(S)\rVert_{\mathrm{op}}=\lVert S\rVert_{\mathrm{op}}$ for every $S\in M_{0}$.

2. (Composites)¶ The composite $\pi'\circ\pi:M_{0}\to M_{2}$ is a trace-preserving embedding of $(H_{0},M_{0},\Omega_{0})$ into $(H_{2},M_{2},\Omega_{2})$, and $V_{\pi'\circ\pi}=V_{\pi'}V_{\pi}$.