Embedding Crossed Products into a Unital Simple AF-algebra

Abstract

Let $X$ be a compact metric space and let $\af$ be a homeomorphism on $X.$ Related to a theorem of Pimsner, we show that $C(X)\rtimes_{\af}\Z$ can be embedded into a unital simple AF-algebra if and only if there is a strictly positive $\af$-invariant Borel probability measure. Suppose that $\Lambda$ is a $\Z^d$ action on $X.$ If $C(X)\rtimes_{\Lambda}\Z$ can be embedded into a unital simple AF-algebra, then there must exist a strictly positive $\Lambda$-invariant Borel probability measure. We show that, if in addition, there is a generator $\af_1$ of $\Lambda$ such that $(X, \af_1)$ is minimal and unique ergodic, then $C(X)\rtimes_{\Lambda}\Z^d$ can be embedded into a unital simple AF-algebra with a unique tracial state. Let $A$ be a unital separable amenable simple \CA 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 $\Lambda: G\to Aut(A)$ is a \hm. Then $A\rtimes_{\Lambda} G$ can always be embedded into a unital simple AF-algebra.