HAAGERUP APPROXIMATION PROPERTY FOR QUANTUM REFLECTION GROUPS

In this paper we prove that the duals of the quantum reflection groups H-N(S+) have the Haagerup property for all N >= 4 and s is an element of [1, infinity). We use the canonical arrow pi : C(H-N(S+)) -> 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 G of Kac type.