Let $M$ be a semi-finite von Neumann algebra with a faithful normal semi-finite trace $\tau$. Let $\mathcal A$ be an ultraweakly closed subset of $M$ which is bounded in both operator norm and the $L^2$-norm. Then $\mathcal A$ is closed on $L^2(M,\tau).$
Here $L^2(M,\tau)$ is the standard representation of $(M,\tau).$ Let try: First I consider the inclusion map $\iota: M\to L^2(M,\tau)$. If I show that $\iota$ is ultraweak-weak continuous then $\iota(\mathcal A)$ will be bounded and weakly closed in $L^2(M,\tau)$. This is my idea. But I am not sure that will work and also I am unable to show that $\iota$ is ultraweakly-weak continuous. Please help me to solve this. Any help is appreciated.
The map $ι$ is ultraweakly continuous by definition of Haagerup's standard form. Indeed, the latter is by definition a von Neumann subalgebra of $\def\B{{\rm B}}\B(H)$ for a certain Hilbert space $H$ (namely, $\def\L{{\rm L}}H=\L^2(M)$), and von Neumann subalgebras of $\B(H)$ are automatically ultraweakly closed.
As a side remark, the trace $τ$ is irrelevant here, since the standard form $\L^2(M)$ does not depend on $τ$.