Subalgebras of simple AF-algebras

Abstract

It is shown that if $A$ is a separable, exact $C^*$-algebra which satisfies the Universal Coefficient Theorem (UCT) and has a faithful, amenable trace, then A admits a trace-preserving embedding into a simple, unital AF-algebra with a unique trace. Modulo the UCT, this provides an abstract characterization of $C^*$-sub­algebras of simple, unital AF-algebras.

As a consequence, for a countable, discrete, amenable group $G$ acting on a second countable, locally compact, Hausdorff space $X$, $C_0(X) \rtimes_r G$ embeds into a simple, unital AF-algebra if, and only if, $X$ admits a faithful, invariant, Borel, probability measure. Also, for any countable, discrete, amenable group $G$, the reduced group $C^*$-algebra $C^*_r(G)$ admits a trace-preserving embedding into the universal UHF-algebra.