Michael's selection theorem and applications to the Maréchal topology

Abstract

The Mar\'echal topology, also called the Effros-Mar\'echal topology, is a natural topology one can put on the space of all von Neumann subalgebras of a given von Neumann algebra. It is a result of Mar\'echal from 1973 that this topology is Polish as soon as the ambient algebra has separable predual, but the sketch of proof in her research announcement appears to have a small gap. Our main goal in this paper is to fill this gap by a careful look at the topologies one can put on the space of weak-$*$ closed subspaces of a dual space. We also indicate how Michael's selection theorem can be used as a step towards Mar\'echal's theorem, and how it simplifies the proof of an important selection result of Haagerup and Winsl{\o}w for the Mar\'echal topology. As an application, we show that the space of finite von Neumann algebras is $\mathbf\Pi^0_3$-complete.