Fixed-points in the cone of traces on a C*-algebra

TL;DR

This paper explores the existence of invariant traces on C*-algebras under group actions with the fixed-point property for cones, linking group properties to trace existence on Roe algebras.

Contribution

It connects Monod's fixed-point property for cones with the existence of invariant traces on non-unital C*-algebras, especially in the context of Roe algebras.

Findings

Invariant traces exist for certain group actions on C*-algebras.

Non-existence results for traces on Roe algebras under specific conditions.

Provides criteria linking group properties to trace existence on operator algebras.

Abstract

Nicolas Monod introduced the class of groups with the fixed-point property for cones, characterized by always admitting a non-zero fixed point whenever acting (suitably) on proper weakly complete cones. He proved that his class of groups contains the class of groups with subexponential growth and is contained in the class of supramenable groups. In this paper we investigate what Monod's results say about the existence of invariant traces on (typically non-unital) C*-algebras equipped with an action of a group with the fixed-point property for cones. As an application of these results we provide results on the existence (and non-existence) of traces on the (non-uniform) Roe algebra.