Mathematische Annalen

, Volume 357, Issue 1, pp 355–400 | Cite as

The \(K\)-theory of free quantum groups

  • Roland Vergnioux
  • Christian VoigtEmail author


In this paper we study the \(K\)-theory of free quantum groups in the sense of Wang and Van Daele, more precisely, of free products of free unitary and free orthogonal quantum groups. We show that these quantum groups are \(K\)-amenable and establish an analogue of the Pimsner–Voiculescu exact sequence. As a consequence, we obtain in particular an explicit computation of the \(K\)-theory of free quantum groups. Our approach relies on a generalization of methods from the Baum–Connes conjecture to the framework of discrete quantum groups. This is based on the categorical reformulation of the Baum–Connes conjecture developed by Meyer and Nest. As a main result we show that free quantum groups have a \(\gamma \)-element and that \(\gamma = 1\). As an important ingredient in the proof we adapt the Dirac-dual Dirac method for groups acting on trees to the quantum case. We use this to extend some permanence properties of the Baum–Connes conjecture to our setting.

Mathematics Subject Classification (2000)

20G42 46L80 19K35 



We would like to thank G. Skandalis for helpful comments. The first named author was supported by ANR grant ANR-09-BLAN-0059 AGORA. The second named author was supported by DFG (SFB 878).


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques Nicolas OresmeUniversité de Caen Basse-NormandieCaen CedexFrance
  2. 2.Mathematisches InstitutWestfälische Wilhelms-Universität MünsterMünsterGermany

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