AF-embedding of the crossed products of AH-algebras by finitely generated abelian groups

Abstract

Let $X$ be a compact metric space and let $Λ$ be a $\Z^k$ ($k\ge 1$) action on $X.$ We give a solution to a version of Voiculescu's problem of AF-embedding: The crossed product $C(X)\rtimes_Λ\Z^k$ can be embedded into a unital simple AF-algebra if and only if $X$ admits a strictly positive $Λ$-invariant Borel probability measure. Let $C$ be a unital AH-algebra, let $G$ be a finitely generated abelian group and let $Λ: G\to Aut(C)$ be a monomorphism. We show that $C\rtimes_Λ G$ can be embedded into a unital simple AF-algebra if and only if $C$ admits a faithful $Λ$-invariant tracial state.