Crossed-products by locally compact groups: Intermediate subfactors

TL;DR

This paper investigates actions of locally compact groups on von Neumann factors, establishing conditions for irreducibility and describing intermediate subfactors in the totally disconnected case, thus generalizing previous results and highlighting limitations for non-discrete groups.

Contribution

It provides new sufficient conditions for irreducibility and describes intermediate subfactors for totally disconnected groups, extending prior work to a broader class of groups.

Findings

Conditions ensuring irreducibility of crossed-product inclusions.

Characterization of intermediate subfactors as crossed-products by closed subgroups.

Limitations of previous strategies for non-discrete groups.

Abstract

We study actions of locally compact groups on von Neumann factors and the associated crossed-product von Neumann algebras. In the setting of totally disconnected groups we provide sufficient conditions on an action $G\curvearrowright Q$ ensuring that the inclusion $Q \subset Q \rtimes G$ is irreducible and that every intermediate subfactor is of the form $Q \rtimes H$ for a closed subgroup $H<G$. This partially generalizes a result of Izumi-Longo-Popa [ILP98] and Choda [Ch78]. We moreover show that one can not hope to use their strategy for non-discrete groups.