On locally finite dimensional traces

TL;DR

This paper investigates approximation properties of traces on simple C*-algebras, showing that locally finite dimensional traces form a convex set and are uniformly LFD on locally reflexive algebras, with specific results for reduced group C*-algebras.

Contribution

It provides new insights into the structure of traces on simple C*-algebras, including convexity and uniform LFD properties, and establishes that reduced C*-algebras of amenable ICC groups are strong-NF.

Findings

Locally finite dimensional traces form a convex set on simple C*-algebras.

Locally finite dimensional traces are automatically uniformly LFD on locally reflexive C*-algebras.

All traces on reduced C*-algebras of amenable ICC groups are uniformly LFD.

Abstract

We partially resolve three open questions on approximation properties of traces on simple C*-algebras. We partially answer two questions raised by Nate Brown by showing that locally finite dimensional (LFD) traces form a convex set on simple C*-algebras and that they are automatically uniformly LFD on locally reflexive C*-algebras. We prove that all the traces on the reduced \C-algebra $C^*_r(\Gamma)$ of a discrete amenable ICC group $\Gamma$ are uniformly LFD, and conclude that $C^*_r(\Gamma)$ is strong-NF in the sense of Blackadar-Kirchberg in this case. This partially answers another open question raised by Brown.