Subalgebras of simple AF-algebras

TL;DR

This paper characterizes when certain separable, exact C*-algebras with a faithful, amenable trace can be embedded into simple AF-algebras, linking algebraic properties to group actions and invariant measures.

Contribution

It provides an abstract characterization of subalgebras of simple AF-algebras under the UCT, and establishes embedding results for crossed products and group C*-algebras.

Findings

C*-algebras with faithful, amenable traces embed into simple AF-algebras

Crossed products by amenable groups embed into AF-algebras if X admits an invariant measure

Reduced group C*-algebras embed into the universal UHF-algebra

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 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.