Inductive Limits of Tracial W*-Probability Spaces along Trace-Preserving Embeddings

In the setting of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, for every $k\in\mathbb{N}$ let $(H_{k},M_{k},\Omega_{k})$ be a tracial W*-probability space and let $\pi_{k}$ be a trace-preserving embedding of $(H_{k},M_{k},\Omega_{k})$ into $(H_{k+1},M_{k+1},\Omega_{k+1})$. Implementing isometries $V_{\pi}$ of trace-preserving embeddings $\pi$ are those of Trace-Preserving Embeddings of Tracial W*-Probability Spaces, Their Implementing Isometries and Conditional Expectations §isometry, the letter $V$ denoting an operator, not an inner product space as in Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation §spaces; and $\mathcal{S}''$ is the double commutant of a set $\mathcal{S}$ of bounded operators on a complex Hilbert space. Then there are a tracial W*-probability space $(H,M,\Omega)$ and, for every $k\in\mathbb{N}$, a trace-preserving embedding $\rho_{k}$ of $(H_{k},M_{k},\Omega_{k})$ into $(H,M,\Omega)$, with the following properties.

1. (Compatibility)¶ For every $k\in\mathbb{N}$: $\rho_{k+1}(\pi_{k}(S))=\rho_{k}(S)$ for every $S\in M_{k}$, and $V_{\rho_{k+1}}V_{\pi_{k}}=V_{\rho_{k}}$.

2. (Density)¶ The set $\{\rho_{k}(S)\Omega:\ k\in\mathbb{N},\ S\in M_{k}\}$ is dense in $H$.

3. (Generation)¶ $M=\mathcal{A}''$, where $\mathcal{A}=\{\rho_{k}(S):\ k\in\mathbb{N},\ S\in M_{k}\}$.