Examples of weakly amenable discrete quantum groups

We prove that the free orthogonal and free unitary quantum groups FON+ and FUN+ 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.