Connes' metric for states in group algebras

TL;DR

This paper explores Connes' metric on the state space of group C*-algebras, establishing links between word-length growth and topological properties of the state space for discrete groups.

Contribution

It provides new insights into the relationship between word-length growth and the metric topology on the state space of reduced group C*-algebras.

Findings

Established equivalences between word-length growth and topological properties

Connected the degree of group extensions with metric properties

Demonstrated the topological equivalence of two topologies in the state space

Abstract

In this article we follow the main idea of A. Connes for the construction of a metric in the state space of a C*-algebra. We focus in the reduced algebra of a discrete group $\Gamma$, and prove some equivalences and relations between two central objects of this category: the word-length growth (connected with the degree of the extension of $\Gamma$ when the group is an extension of Z by a finite group), and the topological equivalence between the w*-topology and the one introduced with this metric in the state space of $C_r*(\Gamma)$.