Graphs of quantum groups and K-amenability

Pierre Fima (1) , Amaury Freslon (1)

1 IMJ - Institut de Mathématiques de Jussieu

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 qantum 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.

Mots clés: Graphs of C* algebras; K-amenability; quantum groups

Domaines: Algèbres d'opérateurs [math.OA]; K-théorie et homologie [math.KT]; Algèbres quantiques [math.QA]

Pierre Fima, Amaury Freslon. Graphs of quantum groups and K-amenability. 2013. ⟨hal-00846840⟩