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.