We classify torsion actions of free wreath products of arbitrary compact quantum groups and use this to prove that if G is a torsion-free compact quantum group satisfying the strong Baum-Connes property, then G ≀ * S + N also satisfies the strong Baum-Connes property. We then compute the K-theory of free wreath products of classical and quantum free groups by SOq(3).

The well known "associativity property" of the crossed product by a semi-direct product of discrete groups is generalized into the context of discrete quantum groups. This decomposition allows to define an appropriate triangulated functor relating the Baum-Connes property for the quantum semi-direct product to the Baum-Connes property for the discrete quantum groups involved in the construction. The corresponding stability result for the Baum-Connes property generalizes the result [5] of J. Chabert for a quantum semi-direct product under torsion-freeness assumption. The K-amenability connexion between the discrete quantum groups involved in the construction is investigated as well as the torsion phenomena. The analogous strategy can be applied for the dual of a quantum direct product. In this case, we obtain, in addition, a connection with the Künneth formula, which is the quantum counterpart to the result [7] of J. Chabert, S. Echterhoff and H. Oyono-Oyono. Again the K-amenability connexion between the discrete quantum groups involved in the construction is investigated as well as the torsion phenomena.

We study the theory of projective representations for a compact quantum group G, i.e. actions of G on BpHq for some Hilbert space H. We show that any such projective representation is inner, and is hence induced by an Ω-twisted representation for some unitary measurable 2-cocycle Ω on G. We show that a projective representation is continuous, i.e. restricts to an action on the compact operators KpHq, if and only if the associated 2-cocycle is regular, and that this condition is automatically satisfied if G is of Kac type. This allows in particular to characterise the torsion of projective type of p G in terms of the projective representation theory of G. For a given regular unitary 2-cocycle Ω, we then study Ω-twisted actions on C ˚-algebras. We define deformed crossed products with respect to Ω, obtaining a twisted version of the Baaj-Skandalis duality and a quantum version of the Packer-Raeburn's trick. As an application, we provide a twisted version of the Green-Julg isomorphism and obtain the quantum Baum-Connes assembly map for permutation torsion-free discrete quantum groups.