Continuity of operator functions in the topology of local convergence in measure

Abstract

Let a von Neumann algebra M of operators act on a Hilbert space H , τ be a faithful normal semifinite trace on M. Let tτl be the topology of τ-local convergence in measure on the *-algebra S(M,τ) of all τ -measurable operators. We prove the tτl-continuity of the involution on the set of all normal operators in S(M,τ). We investigate the tτl -continuity of operator functions on S(M,τ) . We show that the mapping A↦|A| is tτl-continuous on the set of all partial isometries in M.