AF-embedding of crossed products of AH-algebras by Z and asymptotic AF-embedding

Let A be a unital AH-algebra and let α ∈ Aut(A) be an automorphism. A necessary condition for A ⋊α Z being embedded into a unital simple AF-algebra is the existence of a faithful tracial state. If in addition, there is an automorphism κ with κ∗1 = −idK1(A) such that α ◦ κ and κ ◦ α are asymptotically unitarily equivalent, then A⋊α Z can be embedded into a unital simple AF-algebra. Consequently, in the case that A is a unital AH-algebra (not necessarily simple) with torsion K1(A), A ⋊α Z can be embedded into a unital simple AF-algebra if and only if A admits a faithful α-invariant tracial state. We also show that if A is a unital AT-algebra then A ⋊α Z can be embedded into a unital simple AF-algebra if and only if A admits a faithful α-invariant tracial state. If X is a compact metric space and Λ : Z → Aut(C(X)) is a homomorphism then C(X)⋊Λ Z 2 can be asymptotically embedded into a unital simple AF-algebra provided that X admits a strictly positive Λ-invariant probability measure. Consequently C(X) ⋊Λ Z 2 is quasidiagonal if X admits a strictly positive Λ-invariant Borel probability measure.