Local Convergence in Measure on Semifinite von Neumann Algebras

Suppose that $\mathcal M$ is a von Neumann algebra of operators on a Hilbert space $\mathcal H$ and $\tau $ is a faithful normal semifinite trace on $\mathcal M$. The set $\widetilde {\mathcal M}$ of all $\tau $-measurable operators with the topology $t_{\tau }$ of convergence in measure is a topological $*$-algebra. The topologies of $\tau $-local and weakly $\tau $-local convergence in measure are obtained by localizing $t_{\tau }$ and are denoted by $t_{\tau \mathrm l}$ and $t_{\mathrm w\tau \mathrm l}$, respectively. The set $\widetilde {\mathcal M}$ with any of these topologies is a topological vector space. The continuity of certain operations and the closedness of certain classes of operators in $\widetilde {\mathcal M}$ with respect to the topologies $t_{\tau \mathrm l}$ and $t_{\mathrm w\tau \mathrm l}$ are proved. S.M. Nikol'skii's theorem (1943) is extended from the algebra $\mathcal B(\mathcal H)$ to semifinite von Neumann algebras. The following theorem is proved: {\itshape For a von Neumann algebra $\mathcal M$ with a faithful normal semifinite trace $\tau $, the following conditions are equivalent\textup : \textup {(i)} the algebra $\mathcal M$ is finite\textup ; \textup {(ii)} $t_{\mathrm w\tau \mathrm l}= t_{\tau \mathrm l}$\textup ; \textup {(iii)} the multiplication is jointly $t_{\tau \mathrm l}$-continuous from $\widetilde {\mathcal M}\times \widetilde {\mathcal M}$ to $\widetilde {\mathcal M}$\textup ; \textup {(iv)} the multiplication is jointly $t_{\mathrm w\tau \mathrm l}$-continuous from $\widetilde {\mathcal M}\times \widetilde {\mathcal M}$ to $\widetilde {\mathcal M}$\textup ; \textup {(v)} the involution is $t_{\tau \mathrm l}$-continuous from $\widetilde {\mathcal M}$ to $\widetilde {\mathcal M}$.}