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Graphs of quantum groups and \(K\)-amenability.(English)Zbl 1297.46048

Summary: Building on a construction of J.-P. Serre, we associate to any graph of \(C^\ast\)-algebras a maximal and a reduced fundamental \(C^\ast\)-algebra and use this theory to construct the fundamental quantum group of a graph of discrete quantum groups. This construction naturally gives rise to a quantum Bass-Serre tree which can be used to study the \(K\)-theory of the fundamental quantum group. To illustrate the properties of this construction, we prove that, if all the vertex quantum groups are amenable, then the fundamental quantum group is \(K\)-amenable. This generalizes previous results ofP. Julg andA. Valette [J. Funct. Anal. 58, 194–215 (1984;Zbl 0559.46030)],R. Vergnioux [ibid. 212, No. 1, 206–221 (2004;Zbl 1064.46064)] and the first author [ibid. 265, No. 4, 507–519 (2013;Zbl 1316.46044)]. Our proof, even for classical groups, is quite different from the original proof of Julg and Valette, which does not seem to extend straightforwardly to the quantum setting.

MSC:

46L65 Quantizations, deformations for selfadjoint operator algebras
46L09 Free products of \(C^*\)-algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations

Cite

References:

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This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
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