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

Abstract

Let $G$ be a discrete group acting on a von Neumann algebra $M$ by properly outer $\ast$-automorphisms. In this paper we study the containment $M\subseteq M{⋊}_{\alpha }G$ 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 $w^{\ast }$-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 $w^{\ast }$-continuous maps on $M$-bimodules to $\ast$-automorphisms of the containing von Neumann algebras.

27 pages. Minor revisions after referee's comments. Internat. J. Math., to appear

References in corpus (2)

• Bimodules over Cartan MASAs in von Neumann Algebras, Norming Algebras, and Mercer's Theorem
• Conditional expectations onto maximal abelian *-subalgebras

Cited by in corpus (1)

• Crossed-products by locally compact groups: Intermediate subfactors