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

Abstract

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 \(\mathcal M\).