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

Abstract

We study the theory of projective representations for a compact quantum group $\mathbb{G}$, i.e. actions of $\mathbb{G}$ on $\mathcal{B}(H)$ for some Hilbert space $H$. We show that any such projective representation is inner, and is hence induced by an $\Omega$-twisted representation for some unitary measurable $2$-cocycle $\Omega$ on $\mathbb{G}$. We show that a projective representation is continuous, i.e. restricts to an action on the compact operators $\mathcal{K}(H)$, if and only if the associated $2$-cocycle is regular, and that this condition is automatically satisfied if $\mathbb{G}$ is of Kac type. This allows in particular to characterise the torsion of projective type of $\widehat{\mathbb{G}}$ in terms of the projective representation theory of $\mathbb{G}$. For a given regular unitary $2$-cocycle $\Omega$, we then study $\Omega$-twisted actions on C$^*$-algebras. We define deformed crossed products with respect to $\Omega$, 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.