Minimality of convergence in measure topologies on finite von Neumann algebras

Abstract

We prove that the natural embedding of the metric ideal space on a finite von Neumann algebra M into the *-algebra of measurable operators M̃ endowed with the topology of convergence in measure is continuous. Using this fact, we prove that the topology of convergence in measure is a minimal one among all metrizable topologies consistent with the ring structure on M̃.