The Baum-Connes conjecture for free orthogonal quantum groups

Voigt, C. (2011) The Baum-Connes conjecture for free orthogonal quantum groups. Advances in Mathematics, 227(5), pp. 1873-1913. (doi: 10.1016/j.aim.2011.04.008)



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We prove an analogue of the Baum–Connes conjecture for free orthogonal quantum groups. More precisely, we show that these quantum groups have a γ-element and that γ=1. It follows that free orthogonal quantum groups are K-amenable. We compute explicitly their K-theory and deduce in the unimodular case that the corresponding reduced C<sup>⁎</sup>-algebras do not contain nontrivial idempotents. Our approach is based on the reformulation of the Baum–Connes conjecture by Meyer and Nest using the language of triangulated categories. An important ingredient is the theory of monoidal equivalence of compact quantum groups developed by Bichon, De Rijdt and Vaes. This allows us to study the problem in terms of the quantum group SUq(2). The crucial part of the argument is a detailed analysis of the equivariant Kasparov theory of the standard Podleś sphere.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Voigt, Dr Christian
Authors: Voigt, C.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Advances in Mathematics
Journal Abbr.:Adv. Math.
ISSN (Online):1090-2082
Copyright Holders:Copyright © 2011 Elsevier
First Published:First published in Advances in Mathematics 2011 227(5):1873-1913
Publisher Policy:Reproduced in accordance with the copyright policy of the publisher

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