Examples of weakly amenable discrete quantum groups

TL;DR

This paper demonstrates that certain free orthogonal and unitary quantum groups are weakly amenable with Cowling-Haagerup constant 1, extending results to quantum automorphism groups and non-unimodular discrete quantum groups.

Contribution

It proves weak amenability and computes Cowling-Haagerup constants for specific quantum groups, extending to automorphism groups and non-unimodular cases.

Findings

Wang's free orthogonal and free unitary quantum groups are weakly amenable.

Cowling-Haagerup constant is equal to 1 for these quantum groups.

Results extend to quantum automorphism groups and some non-unimodular discrete quantum groups.

Abstract

We prove that Wang's free orthogonal and free unitary quantum groups are weakly amenable and that their Cowling-Haagerup constant is equal to 1. This is achieved by estimating the completely bounded norm of the projections on the coefficients of irreducible representations of their compact duals. An argument of monoidal equivalence then allows us to extend this result to quantum automorphism groups of finite spaces and even yields some examples of weakly amenable non-unimodular discrete quantum groups with the Haagerup property.