The Baum–Connes conjecture for free orthogonal quantum groups

By Christian Voigt. 2009

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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 ∗ -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 Podle´

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e1777ed12d76b1e1f672c99787f8acd920fa314a

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Citations in the corpus

On the classification of easy quantum groups (2012, methodology): “cribed above. For a direct computation, use that t k 1= √1 n , k = 1,...,n, and ∑n =1tki = √ n∑n k= tkitk1 = 0 for all i ≠ 1. These isomorphism of C∗-algebras enable us to transfer a result of Voigt ([V11]) on the K-theory of O+ n to the quantum groups B+n, B′ n + and B#+ n . The K-theory of S+ n, S”
The Haagerup property for locally compact quantum groups (2013, background): “It was shown in [75] that the duals of free orthogonal groups satisfy an appropriate version of the Baum–Connes conjecture.”
CCAP for Universal Discrete Quantum Groups (2013, methodology): “(1)One should note that this parametrization of πz is different from the one of [41] by z ↔ 1 − z.”
Noncommutative K\"ahler Structures on Quantum Homogeneous Spaces (2011, background): “It is motivated by the appearance of noncommutative complex structures in a number of areas of noncommutative geometry, such as the construction of spectral triples for quantum groups [20, 7, 2], geometric representation theory for quantum groups [16, 25, 17, 18], the interaction of noncommutative g”