MF traces and the Cuntz semigroup

TL;DR

This paper investigates the properties of traces on C*-algebras, especially their relation to the Cuntz semigroup and MF traces, with applications to crossed products and classification of nuclear C*-algebras.

Contribution

It establishes the existence of state-preserving morphisms from the Cuntz semigroup of a C*-algebra to that of the ultrapower of the universal UHF-algebra, and applies this to classify when crossed products are MF.

Findings

Every trace on certain C*-algebras induces a state-preserving morphism on the Cuntz semigroup.

All traces on the reduced crossed product of an AI-algebra by a free group are MF.

Characterization of when crossed products of specific nuclear C*-algebras are MF.

Abstract

A trace $\tau$ on a separable C*-algebra $A$ is called matricial field (MF) if there is a trace-preserving morphism from $A$ to $Q_\omega$, where $Q_\omega$ denotes the norm ultrapower of the universal UHF-algebra $Q$. In general, the trace $\tau$ induces a state on the Cuntz semigroup $Cu(A)$. We show there is always a state-preserving morphism from $Cu(A)$ to $Cu(Q_\omega)$. As an application, if $A$ is an AI-algebra and $F$ is a free group acting on $A$, then every trace on the reduced crossed product $A \rtimes F$ is MF. This further implies the same result when $A$ is an AH-algebra with the ideal property such that $K_1(A)$ is a torsion group. We also use this to characterize when $A \rtimes F$ is MF (i.e. admits an isometric morphism into $Q_\omega$) for many simple, nuclear C*-algebras $A$.