When weak and local measure convergence implies norm convergence

Abstract

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 t τ 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 I B = { A ∈ S ( M , τ ) h : − B ≤ A ≤ B } and K B = { A ∈ S ( M , τ ) : A ⁎ A ≤ B } are convex and t τ -closed in S ( M , τ ) . In this case, we have I B = { B T B : T ∈ M h and ‖ T ‖ ≤ 1 } and, for invertible B , we describe the set of extreme points of the set I B . 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 I B is t τ -compact. The t τ -compactness of I B for all τ -compact operators B characterizes these algebras.