Noncommutative Vitali-Hahn-Saks Theorem holds precisely for finite $W^{\ast }$-algebras

Abstract

It is shown that the bona fide generalization of the Vitali-Hahn-Saks Theorem to von Neumann algebras is possible if, and only if, the algebra is finite. This settles the problem on the noncommutative Vitali-Hahn-Saks Theorem completely and provides new means of characterizing finite von Neumann algebras.

8 pages