# Traces on reduced group C*-algebras

## TL;DR

This paper characterizes when the reduced group C*-algebra admits a non-zero trace, linking it precisely to the openness of the group's amenable radical, thus resolving a previously posed question.

## Contribution

It provides a complete characterization of the existence of non-zero traces on reduced group C*-algebras in terms of the group's amenable radical being open.

## Key findings

- Reduced group C*-algebra admits a non-zero trace iff the amenable radical is open.
- Answers a question posed by Forrest, Spronk, and Wiersma.
- Clarifies the relationship between group properties and C*-algebra traces.

## Abstract

In this short note we prove that the reduced group C*-algebra of a locally compact group admits a non-zero trace if and only if the amenable radical of the group is open. This completely answers a question raised by Forrest, Spronk and Wiersma.