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 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.

Untitled

“…That is, if we have a coaction of (𝐴, Δ) on ℬ(𝐻), and 𝐵 ⊆ ℬ(𝐻) with a quantum graph 𝑆 ⊆ ℬ(𝐻), is it possible to place conditions on the coaction so as we obtain a coaction on 𝑆 (whatever this might mean, at this level of generality). For more on coactions on ℬ(𝐻) see, for example, [20,Section 3].…”

Quantum graphs: different perspectives, homomorphisms and quantum automorphisms

“…That is, if we have a coaction of (A, ∆) on B(H), and B ⊆ B(H) with a quantum graph S ⊆ B(H), is it possible to place conditions on the coaction so as we obtain a coaction on S (whatever this might mean, at this level of generality). For more on coactions on B(H) see, for example, [18, Section 3].…”