Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map

We study the theory of projective representations for a compact quantum group G, i.e. actions of G on B(H) 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 K(H), 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 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.