Graphs of quantum groups and K-amenability

Building on a construction of J.-P. Serre, we associate to any graph of C⁎-algebras a maximal and a reduced fundamental C⁎-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 of P. Julg, A. Valette, R. Vergnioux and the first author. 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.