Group C*-Algebras: amenability and nuclearity

Abstract

The goal of this work is to study group $C^*$-algebras, with particular emphasis on the notions of amenability and nuclearity. We develop the necessary background on representations of groups and Banach $*$-algebras, in order to construct two $C^*$-algebras associated to a locally compact group. This objects will allow us to investigate the interplay between algebraic properties of groups and analytic properties of operator algebras.

The concepts of amenability and nuclearity are then introduced. The first describes a fundamental regularity property of groups, admitting several equivalent characterizations. The second is a regularity property of $C^*$-algebras, expressed through the behavior of tensor products and finite-dimensional approximations. These two notions, originating in seemingly distinct mathematical settings, are shown to be equivalent in the culminating theorem of this work.