Haagerup approximation property via bimodules

TL;DR

This paper explores the Haagerup approximation property (HAP) for von Neumann algebras, establishing that quantum group von Neumann algebras with HAP also possess the property, extending known results to more general settings.

Contribution

It proves that von Neumann algebras of locally compact quantum groups with HAP also have the HAP, filling a gap in the theory and extending results to quantum groups.

Findings

Quantum group von Neumann algebras with HAP have the HAP

Extension of HAP results to infinite and quantum group settings

New insights into the structure of von Neumann algebras with HAP

Abstract

The Haagerup approximation property (HAP) is defined for finite von Neumann algebras in such a way that the group von Neumann algebra of a discrete group has the HAP if and only if the group itself has the Haagerup property. The HAP has been studied extensively for finite von Neumann algebras and it is recently generalized for arbitrary von Neumann algebras by Caspers-Skalski and Okayasu-Tomatsu. One of the motivations behind the generalization is the fact that quantum group von Neumann algebras are often infinite even though the Haagerup property has been defined successfully for locally compact quantum groups by Daws-Fima-Skalski-White. In this paper, we partly fill this gap by proving that the von Neumann algebra of a locally compact quantum group with the Haagerup property has the HAP. This is new even for genuine locally compact groups.