CCAP for universal discrete quantum groups

TL;DR

This paper proves that the discrete duals of certain free quantum groups possess the Haagerup property and CCAP, extending previous results to a broader class of quantum groups using monoidal equivalence and free product techniques.

Contribution

It establishes the Haagerup property and CCAP for the discrete duals of free orthogonal, unitary, and automorphism quantum groups, generalizing prior Kac algebra results.

Findings

Discrete duals of free orthogonal quantum groups have Haagerup property.

Discrete duals of free orthogonal quantum groups have CCAP.

Results extend to free unitary and automorphism quantum groups.

Abstract

We show that the discrete duals of the free orthogonal quantum groups have the Haagerup property and the completely contractive approximation property. Analogous results hold for the free unitary quantum groups and the quantum automorphism groups of finite-dimensional C*-algebras. The proof relies on the monoidal equivalence between free orthogonal quantum groups and SUq(2) quantum groups, on the construction of a sufficient supply of bounded central functionals for SUq(2) quantum groups, and on the free product techniques of Ricard and Xu. Our results generalize previous work in the Kac setting due to Brannan on the Haagerup property, and due to the second author on the CCAP.