Graphs of quantum groups and K-amenability

TL;DR

This paper develops a framework for associating quantum groups to graphs of C*-algebras, enabling the study of their K-theory and proving K-amenability under certain conditions, thus extending classical results to quantum settings.

Contribution

It introduces a novel construction of fundamental quantum groups from graphs of discrete quantum groups, generalizing classical Bass-Serre theory to quantum groups.

Findings

If all vertex quantum groups are amenable, then the fundamental quantum group is K-amenable.

The construction yields a quantum Bass-Serre tree useful for K-theory analysis.

Extends classical K-amenability results to the quantum group context.

Abstract

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.