Crossed products of UHF algebras by some amenable groups

Let A be a UHF C*-algebra. It is shown that for every homo morphism α:Zⁿ⟶ Aut(A) there exists an AF embedding ρ: A ⋊_ρZⁿ⟶B such that ρ*:K₀(A ⋊_ρZⁿ⟶K₀(B) is also injective. Using Green’s imprimitivity theorem it will follow that if A is UHF and α: G ⟶ Aut(A) is a homomorphism then A ⋊_ρG is always quasidiagonal for a large class of amenable groupsincluding all extensions of discrete abelian groups by compact (not necessarily discrete or abelian) groups.