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 \(\mathcal{M}\) into the *-algebra of measurable operators \(\tilde {\mathcal {M}}\) 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 \(\tilde {\mathcal {M}}\).