Relative Haagerup property for arbitrary von Neumann algebras

TL;DR

This paper introduces a new relative Haagerup property for von Neumann algebras, explores its invariance under certain conditions, and applies it to free product constructions, with implications for quantum algebra examples.

Contribution

It defines the relative Haagerup property for arbitrary von Neumann algebras, proves its invariance under finite conditions, and applies it to free products and quantum algebra examples.

Findings

The property is independent of the conditional expectation when the smaller algebra is finite.

The property is stable under free products with finite-dimensional amalgamation.

Examples include q-deformed Hecke-von Neumann algebras and quantum orthogonal groups.

Abstract

We introduce the relative Haagerup approximation property for a unital, expected inclusion of arbitrary von Neumann algebras and show that if the smaller algebra is finite then the notion only depends on the inclusion itself, and not on the choice of the conditional expectation. Several variations of the definition are shown to be equivalent in this case, and in particular the approximating maps can be chosen to be unital and preserving the reference state. The concept is then applied to amalgamated free products of von Neumann algebras and used to deduce that the standard Haagerup property for a von Neumann algebra is stable under taking free products with amalgamation over finite-dimensional subalgebras. The general results are illustrated by examples coming from q-deformed Hecke-von Neumann algebras and von Neumann algebras of quantum orthogonal groups.