AF-embedding of crossed products of AH-algebras by $\mathbb{Z}$ and asymptotic AF-embedding

Abstract

Let $A$ be a unital AH-algebra and let $\alpha \in Aut(A)$ be an automorphism. A necessary condition for $A{⋊}_{\alpha }\mathbb{Z}$ being embedded into a unital simple AF-algebra is the existence of a faithful tracial state. If in addition, there is an automorphism $\kappa$ with ${\kappa }_{\ast 1}=-{\mathrm{id}}_{K_1(A)}$ such that $\alpha \circ \kappa$ and $κ\circ \af$ are asymptotically unitarily equivalent, then $A\rtimes_{\af}\Z$ 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 $K_1(A),$ $A{⋊}_{\alpha }\mathbb{Z}$ can be embedded into a unital simple AF-algebra if and only if $A$ admits a faithful $\alpha$-invariant tracial state. We also show that if $A$ is a unital A$\T$-algebra then $A{⋊}_{\alpha }\mathbb{Z}$ can be embedded into a unital simple AF-algebra if and only if $A$ admits a faithful $\af$-invariant tracial state. If $X$ is a compact metric space and $\Lambda :{\mathbb{Z}}^2\to Aut(C(X))$ is a \hm then $C(X){⋊}_{\Lambda }{\mathbb{Z}}^2$ can be asymptotically embedded into a unital simple AF-algebra provided that $X$ admits a strictly positive $\Lambda$-invariant probability measure. Consequently $C(X){⋊}_{\Lambda }{\mathbb{Z}}^2$ is quasidiagonal if $X$ admits a strictly positive $\Lambda$-invariant Borel probability measure.