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A complete formulation of the Baum-Connes conjecture for the action of discrete quantum groups. (English) Zbl 1051.19003

Replacing locally compact groups by discrete quantum groups, the authors formulate a Baum-Connes conjecture (without coefficients) for discrete quantum groups. If \(G\) is a discrete group, the authors’ conjecture for \(C_{0} (G)\) agrees with the usual Baum-Connes conjecture for \(G\). The authors’ approach runs parallel to the development of the standard Baum-Connes conjecture: they extend to their context the notions of analytic assembly map and classifying space for proper actions.
In particular, for a discrete quantum group \(A\), the domain of the Baum-Connes map is a limit of \(A\)-equivariant \(K\)-homology groups of \(C^{*}\)-algebras modeled on those defined by J. Cuntz [Geom. Funct. Anal. 12, 307–329 (2002; Zbl 1037.46060)]. [The \(A\)-equivariant \(K\)-homology groups are those defined by S. Baaj and G. Skandalis, \(K\)-Theory 2, 683–721 (1989; Zbl 0683.46048).] The authors discuss examples and show that their Baum-Connes conjecture is true for finite-dimensional quantum groups.

MSC:

19K35 Kasparov theory (\(KK\)-theory)
46L80 \(K\)-theory and operator algebras (including cyclic theory)
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
58B32 Geometry of quantum groups
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