Minimality of Convergence in Measure Topologies on Finite von Neumann Algebras

We prove that the natural embedding of the metric ideal space on a finite von Neumann algebra ${\mathscr M}$ into the $*$-algebra of measurable operators $\widetilde {\mathscr M}$ 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 $\widetilde {\mathscr M}$.