Intermediate subalgebras and bimodules for crossed products of general von Neumann algebras

TL;DR

This paper characterizes intermediate von Neumann algebras and bimodules within crossed products of general von Neumann algebras under group actions, extending previous results and providing new insights into their structure and automorphisms.

Contribution

It extends the characterization of intermediate von Neumann algebras and bimodules to the non-factor case, generalizing earlier factor-specific results and applying to broader group actions.

Findings

Characterization of intermediate von Neumann algebras in crossed products.

Description of $M$-bimodules closed in the Bures topology.

Extension of isometric $w^*$-continuous maps to automorphisms.

Abstract

Let $G$ be a discrete group acting on a von Neumann algebra $M$ by properly outer $*$-automorphisms. In this paper we study the containment $M \subseteq M\rtimes_\alpha G$ of $M$ inside the crossed product. We characterize the intermediate von Neumann algebras, extending earlier work of other authors in the factor case. We also determine the $M$-bimodules that are closed in the Bures topology and which coincide with the $w^*$-closed ones under a mild hypothesis on $G$. We use these results to obtain a general version of Mercer's theorem concerning the extension of certain isometric $w^*$-continuous maps on $M$-bimodules to $*$-automorphisms of the containing von Neumann algebras.