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

Let $X$ be a compact metric space and let $\Lambda$ 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 $C(X)\rtimes_{\Lambda}\Z^k$ can be embedded into a unital simple AF-algebra if and only if $X$ admits a strictly positive $\Lambda$-invariant Borel probability measure. Let $C$ be a unital AH-algebra, let $G$ be a finitely generated abelian group and let $\Lambda: G\to Aut(C)$ be a monomorphism. We show that $C\rtimes_{\Lambda} G$ can be embedded into a unital simple AF-algebra if and only if $C$ admits a faithful $\Lambda$-invariant tracial state.