Permanence of approximation properties for discrete quantum groups

TL;DR

This paper investigates how key approximation properties like weak amenability and the Haagerup property are preserved under various constructions in discrete quantum groups, including free products with amalgamation and relative amenability.

Contribution

It introduces new permanence results for these properties, especially in the context of free products with amalgamation and the concept of relative amenability.

Findings

Permanence of weak amenability and Haagerup property under free products with finite quantum subgroup amalgamation.

Introduction of a notion of relative amenability for discrete quantum groups.

Linking relative amenability with amenable equivalence of von Neumann algebras.

Abstract

We prove several results on the permanence of weak amenability and the Haagerup property for discrete quantum groups. In particular, we improve known facts on free products by allowing amalgamation over a finite quantum subgroup. We also define a notion of relative amenability for discrete quantum groups and link it with amenable equivalence of von Neumann algebras, giving additional permanence properties.