Équivalence monoïdale de groupes quantiques et K-théorie bivariante

Saad Baaj; Jonathan Crespo

doi:10.24033/bsmf.2751

paper

Équivalence monoïdale de groupes quantiques et K-théorie bivariante

arXiv:1507.06808 · doi:10.24033/bsmf.2751

Abstract

In this article, we generalize to the case of regular locally compact quantum groups, two important results concerning actions of compact quantum groups. Let $G_{1}$ and $G_{2}$ be two monoidally equivalent regular locally compact quantum groups in the sense of De Commer. We introduce an induction procedure and we build an equivalence of the categories $A^{G_{1}}$ and $A^{G_{2}}$ consisting of continuous actions of $G_{1}$ and $G_{2}$ on $C^{*}$ -algebras. As an application of this result, we derive a canonical equivalence of the categories $K K^{G_{1}}$ and $K K^{G_{2}}$ . We introduce and investigate a notion of actions on $C^{*}$ -algebras of measured quantum groupoids on a finite basis. The proof of the equivalence between $K K^{G_{1}}$ and $K K^{G_{2}}$ relies on a version of the Takesaki-Takai duality theorem for continuous actions on $C^{*}$ -algebras of measured quantum groupoids on a finite basis.

In French. Published in Bulletin de la Société Mathématique de France. Minor changes (corrected typos, added a reference)

Cited by in corpus (1)

Actions of measured quantum groupoids on a finite basis