The Baum-Connes conjecture for free orthogonal quantum groups

Details The Baum-Connes conjecture for free orthogonal quantum groups The Baum-Connes conjecture for free orthogonal quantum groups

2009-11-16

arxiv.org/abs/0911.2999v2

Adv. Math. 227 (2011), 1873 - 1913

Mathematics

Operator Algebras

34 pages, final version

Scientific paper

10.1016/j.aim.2011.04.008

We prove an analogue of the Baum-Connes conjecture for free orthogonal quantum groups. More precisely, we show that these quantum groups have a $ \gamma $-element and that $ \gamma = 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^* $-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 $ SU_q(2) $. The crucial part of the argument is a detailed analysis of the equivariant Kasparov theory of the standard Podle\'s sphere.

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