Connes' metric for states in group algebras

Abstract

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 Г, 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 Г when the group is an extension of Z), and the topological relation between the w*-topology and the one introduced with this metric in the state space of C_r*(Г). Recent studies [Antonescu] of Christensen and Antonescu show that, using a variation of the distance introduced by Connes, these topologies are equivalent if the group is of rapid decay, a concept which is equivalent in discrete groups to the concept of polynomial growth for the word-length (there is an extensive survey by Jolissant [Jol] that settles this equivalence). In this article we prove with elementary techniques, that Connes´ metric is finite and induces a topology which is equivalent to the w* topology in the state space, when the group Г is a finite extension of Z. This is not surprising at all, since M Rieffel recently established [Rieffel2] (with a complete different approach) this equivalence for Г=Z.