Group C*-algebras, metrics and an operator theoretic inequality

TL;DR

This paper investigates metrics on the state space of reduced group C*-algebras induced by length functions, proving boundedness for free non-Abelian groups and exploring new metrics for groups of rapid decay, along with a noncommutative Arzela-Ascoli theorem.

Contribution

It introduces a relaxation of length functions for metric construction and proves boundedness results for free groups, extending the understanding of spectral triples and metrics in noncommutative geometry.

Findings

The metric on free non-Abelian groups' state space is bounded.

Many metrics related to length functions exist for groups of rapid decay.

A noncommutative Arzela-Ascoli Theorem is established.

Abstract

On a discrete group G a length function may implement a spectral triple on the reduced group C*-algebra. Following A. Connes, the Dirac operator of the triple then can induce a metric on the state space of reduced group C*-algebra. Recent studies by M. Rieffel raise several questions with respect to such a metric on the state space. Here it is proven that for a free non Abelian group, the metric on the state space is bounded. Further we propose a relaxation in the way a length function is used in the construction of a metric, and we show that for groups of rapid decay there are many metrics related to a length function which all have all the expected properties. The boundedness result for free groups is based on an estimate of the completely bounded norm of a certain Schur multiplier and on some techniques concerning free groups due to U. Haagerup. At the end we have included a noncommutative version of the Arzela-Ascoli Theorem.