CCAP for Universal Discrete 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.