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

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