Haagerup approximation property for quantum reflection groups

Abstract:

In this paper we prove that the duals of the quantum reflection groups $H_N^{s+}$ have the Haagerup property for all $N\ge 4$ and $s\in [1,\infty )$. We use the canonical arrow $\pi : C(H_N^{s+})\to C(S_N^+)$ onto the quantum permutation groups, and we describe how the characters of $C(H_{N}^{s+})$ behave with respect to this morphism $\pi$ thanks to the description of the fusion rules binding irreducible corepresentations of $C(H_N^{s+})$ as in Banica and Vergnioux, 2009. This allows us to construct states on the central $C^*$-algebra $C(H_N^{s+})_0$ generated by the characters of $C(H_{N}^{s+})$ and to use a fundamental theorem proved by M. Brannan giving a method to construct nets of trace-preserving, normal, unital and completely positive maps on the von Neumann algebra of a compact quantum group $\mathbb {G}$ of Kac type.