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

TL;DR

This paper proves that the noncommutative Vitali-Hahn-Saks Theorem holds exactly for finite von Neumann algebras, offering a complete characterization and new insights into their structure.

Contribution

It establishes that the theorem's generalization is valid only for finite von Neumann algebras, resolving a key open problem in the field.

Findings

The theorem holds if and only if the algebra is finite.

Provides a new characterization of finite von Neumann algebras.

Completes the understanding of the noncommutative Vitali-Hahn-Saks Theorem.

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.