Continuity of Operator Functions in the Topology of Local Convergence in Measure

Let a von Neumann algebra $\mathcal M$ of operators act on a Hilbert space $\mathcal {H}$, and let $\tau $ be a faithful normal semifinite trace on $\mathcal M$. Let $t_{\tau \textup {l}}$ be the topology of $\tau $-local convergence in measure on the *-algebra $S(\mathcal M,\tau )$ of all $\tau $-measurable operators. We prove the $t_{\tau \textup {l}}$-continuity of the involution on the set of all normal operators in $S(\mathcal M,\tau )$, investigate the $t_{\tau \textup {l}}$-continuity of operator functions on $S(\mathcal M,\tau )$, and show that the map $A\mapsto |A|$ is $t_{\tau \textup {l}}$-continuous on the set of all partial isometries in $\mathcal M$.