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

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