Traces on locally compact groups

TL;DR

This paper systematically studies traces on locally compact groups and their C*-algebras, introducing the trace kernel and residually-SIN groups, and explores implications for amenability, property (T), and embeddability into AF algebras.

Contribution

It introduces the class of residually-SIN groups, analyzes trace kernels for various groups, and links traces to properties like amenability, property (T), and embeddability into simple AF algebras.

Findings

Existence of traces on reduced C*-algebras is characterized for compactly generated groups.

Unique traces are shown to exist on certain non-discrete groups.

Amenability is equivalent to nuclearity and trace existence in reduced C*-algebras.

Abstract

We conduct a systematic study of traces on locally compact groups, in particular traces on their universal and reduced C*-algebras. We introduce the trace kernel, and examine its relation to the von Neumann kernel and to small-invariant neighbourhood (SIN) quotients. In doing so, we introduce the class of residually-$SIN$ groups, which contains both $SIN$ and maximally almost periodic groups. We examine in detail the trace kernel for connected groups. We study traces on reduced C*-algebras, giving a simple proof for compactly generated groups that existence of such a trace is equivalent to having an open normal amenable subgroup, and we display non-discrete groups admitting unique trace. We finish by examining amenable traces and the factorization property. We show for property (T) groups that amenable trace kernels coincide with von Neumann kernels. We show for totally disconnected groups that amenable trace separation implies the factorization property. We use amenable traces to give a simple proof that amenability of the group is equivalent to simultaneous nuclearity and possessing a trace of its reduced C*-algebra. As a final application of the results obtained in the paper, we address the embeddability of group C*-algebras into simple AF algebras. As a consequence, if a locally compact group is amenable and tracially separated (trace kernel is trivial), then its reduced C*-algebra is quasi-diagonal.