Title: MF traces and the Cuntz semigroup,
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 $A$.