Embedding Crossed Products into a Unital Simple AF-algebra

Let X be a compact metric space and let α be a homeomorphism on X. Related to a theorem of Pimsner, we show that C(X) ⋊α Z can be embedded into a unital simple AF-algebra if and only if there is a strictly positive α-invariant Borel probability measure. Suppose that Λ is a Z action on X. If C(X)⋊Λ Z can be embedded into a unital simple AF-algebra, then there must exist a strictly positive Λ-invariant Borel probability measure. We show that, if in addition, there is a generator α1 of Λ such that (X,α1) is minimal and unique ergodic, then C(X)⋊Λ Z d can be embedded into a unital simple AF-algebra with a unique tracial state. Let A be a unital separable amenable simple C∗-algebra with tracial rank zero and with a unique tracial state which satisfies the Universal Coefficient Theorem and let G be a finitely generated discrete abelian group. Suppose Λ : G → Aut(A) is a homomorphism. Then A⋊Λ G can always be embedded into a unital simple AF-algebra.