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Let $\mathcal{M}$ be a von Neumann algebra equipped with a faithful normal semifinite trace $\tau$, let $S(\mathcal{M},\tau)$ be the algebra of $\tau$-measurable operators.

Q1: If $\mathcal{B}$ is the subalgebra of $S(\mathcal{M},\tau)$. Whether there exists a von Neumann subalgebra $\mathcal{N}$ of $\mathcal{M}$ such that $S(\mathcal{N},\tau\vert_{\mathcal{N}})=\mathcal{B}$.

We know if $\tau$ is finite, it is right.

Q2: If $\mathcal{N}$ is von Neumann subalgebra of $\mathcal{M}$, what conditions we need to make sure that $S(\mathcal{N},\tau\vert_{\mathcal{N}})$ is a subalgebra of $S(\mathcal{M},\tau)$.

Those are stated simply, however, I do not the answers.

Shaoze Pan
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  • You certainly need some additional properties for Q1, even if $\tau$ is finite. Clearly $\mathcal B$ must be self-adjoint and closed in the measure topology. For Q2, you need that $\tau|_{\mathcal N}$ is semi-finite, and I think that's also sufficient. – MaoWao Apr 17 '23 at 19:10
  • @MaoWao thank you. Could you be more specific? Or you can tell me where I can get it from books or papers. – Shaoze Pan Apr 18 '23 at 14:16
  • Take a look at Takesaki's book,specifically book II. Measurable operators with respect to a trace are treated in Chaper IX. Terp's notes on noncommutative $L^p$ spaces are also a good source. – MaoWao Apr 18 '23 at 16:39
  • @MaoWao Thank you. Can you send me your email? So that we can talk frequently. – Shaoze Pan Apr 19 '23 at 02:43

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