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Pré-Publication, Document De Travail Année : 2024

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

Résumé

We study the theory of measurable 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 measurable projective representation is inner, and is hence induced by an Ω-twisted representation for some 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 Ĝ in terms of the projective representation theory of G. For a given regular 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 the Green-Julg isomorphism, and a quantum version of the Packer-Raeburn's trick.
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Dates et versions

hal-04442264 , version 1 (06-02-2024)

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Kenny de Commer, Rubén Martos, Ryszard Nest. Projective representation theory for compact quantum groups and the quantum Baum-Connes assembly map. 2024. ⟨hal-04442264⟩
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