When weak and local measure convergence implies norm convergence

Abstract

© 2019 Elsevier Inc. Let M be a von Neumann algebra of operators on a Hilbert space H and τ be a faithful normal semifinite trace on M. Let tτ be the measure topology on the ⁎-algebra S(M,τ) of all τ-measurable operators. We prove that for B∈S(M,τ)+ the sets IB={A∈S(M,τ)h:−B≤A≤B} and KB={A∈S(M,τ):A⁎A≤B} are convex and tτ-closed in S(M,τ). In this case, we have IB={BTB:T∈Mhand‖T‖≤1} and, for invertible B, we describe the set of extreme points of the set IB. Let M be an atomic von Neumann algebra. We prove that an operator B∈S(M,τ)+ is τ-compact if and only if the set IB is tτ-compact. The tτ-compactness of IB for all τ-compact operators B characterizes these algebras.