The $$K$$ -theory of free quantum groups
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Mathematische Annalen
The K -theory of free quantum groups Roland Vergnioux · Christian Voigt
Received: 23 April 2012 / Revised: 17 November 2012 / Published online: 12 February 2013 © Springer-Verlag Berlin Heidelberg 2013
Abstract 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 γ -element and that γ = 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
1 Introduction A classical result in the theory of C ∗ -algebras is the computation of the K -theory of the reduced group C ∗ -algebra Cr∗ (Fn ) of the free group on n generators by Pimsner and
R. Vergnioux Laboratoire de Mathématiques Nicolas Oresme, Université de Caen Basse-Normandie, BP 5186, Caen Cedex 14032, France e-mail: [email protected] C. Voigt (B) Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany e-mail: [email protected]
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Voiculescu [24]. Their result resolved in particular Kadison’s problem on the existence of nontrivial projections in these C ∗ -algebras. More generally, Pimsner and Voiculescu established an exact sequence for the K -theory of reduced crossed products by free groups [23,24]. This exact sequence is an important tool in operator K -theory. The K -theory of the full group C ∗ -algebra Cf∗ (Fn ) was calculated before by Cuntz in a simple and elegant way, based on a general formula for the K -theory of free products [10]. Motivated by this, Cuntz introduced the notion of K -amenability for discrete groups and gave a shorter proof of the results of Pimsner and Voiculescu [11]. The fact that free groups are K -amenable expresses in a conceptually clear way that full and reduced crossed products for these groups cannot be distinguished on the level of K -theory. The main aim of this paper is to obtain analogous results for free quantum groups. In fact, in the theory of discrete quantum groups, the rôle of free quantum groups is analogous to the rôle of free groups among classical discrete groups. Roughly speaking, any discrete quantum group can be obtained as a quotient of a free quantum group. Classically, the free group on n generators can be describe
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