Michael's selection theorem and applications to the Mar\'echal topology

TL;DR

This paper clarifies the proof that the Maréchal topology on von Neumann subalgebras is Polish, using Michael's selection theorem to simplify the argument and establish the topology's properties.

Contribution

It fills a gap in the proof of the Maréchal topology's Polish property and demonstrates how Michael's selection theorem can be applied to this context.

Findings

Confirmed the Maréchal topology is Polish for separable predual algebras.

Simplified the proof of key selection results using Michael's theorem.

Showed the space of finite von Neumann algebras is 3-complete.

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.