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 \(t_{\tau\text{l}}\) be the topology of \(\tau\) -local convergence in measure on the *-algebra \(S(\mathcal M,\tau)\) of all \(\tau\) -measurable operators. We prove the \(t_{\tau\text{l}}\) -continuity of the involution on the set of all normal operators in \(S(\mathcal M,\tau)\) , investigate the \(t_{\tau\text{l}}\) -continuity of operator functions on \(S(\mathcal M,\tau)\) , and show that the map \(A\mapsto |A|\) is \(t_{\tau\text{l}}\) -continuous on the set of all partial isometries in \(\mathcal M\) .