Approximation properties for dynamical W*-correspondences

TL;DR

This paper studies approximation properties of equivariant W*-correspondences under quantum group actions, introduces a Fell topology for these structures, and characterizes amenability of group actions on von Neumann algebras.

Contribution

It develops a framework for analyzing equivariant W*-correspondences, including a Fell topology and approximation properties, and links these to amenability of group actions.

Findings

Defined a Fell topology on equivariant correspondences

Characterized amenability of group actions via this framework

Analyzed continuity of natural operations on correspondences

Abstract

Let $\mathbb{G}$ be a locally compact quantum group, and $A,B$ von Neumann algebras on which $\mathbb{G}$ acts. We refer to these as $\mathbb{G}$-dynamical W$^*$-algebras. We make a study of $\mathbb{G}$-equivariant $A$-$B$-correspondences, that is, Hilbert spaces $\mathcal{H}$ with an $A$-$B$-bimodule structure by $*$-preserving normal maps, and equipped with a unitary representation of $\mathbb{G}$ which is equivariant with respect to the above bimodule structure. Such structures are a Hilbert space version of the theory of $\mathbb{G}$-equivariant Hilbert C$^*$-bimodules. We show that there is a well-defined Fell topology on equivariant correspondences, and use this to formulate approximation properties for them. Within this formalism, we then characterize amenability of the action of a locally compact group on a von Neumann algebra, using recent results due to Bearden and Crann. We further consider natural operations on equivariant correspondences such as taking opposites, composites and crossed products, and examine the continuity of these operations with respect to the Fell topology.