IUMJ

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

Authors: Huaxin Lin

Issue: Volume 57 (2008), Issue 2, 891-944

Abstract:

Let $A$ be a unital AH-algebra and let $\alpha \in \text{Aut}(A)$ be an automorphism. A necessary condition for $A \rtimes_{\alpha} \mathbb{Z}$ 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 $\kappa_{*1} = -\mathrm{id}_{K_1(A)}$ such that $\alpha \mathrel{\circ} \kappa$ and $\kappa \mathrel{\circ} \alpha$ are asymptotically unitarily equivalent, then $A \rtimes_{\alpha} \mathbb{Z}$ 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 $K_1(A)$, $A \rtimes_{\alpha} \mathbb{Z}$ 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 $\mathrm{A}\mathbb{T}$-algebra then $A \rtimes_{\alpha} \mathbb{Z}$ can be embedded into a unital simple AF-algebra if and only if $A$ admits a faithful $\alpha$-invariant tracial state. Consequently, for any unital simple $\mathrm{A}\mathbb{T}$-algebra $A$, $A \rtimes_{\alpha} \mathbb{Z}$ can always be embedded into a unital simple AF-algebra. If $X$ is a compact metric space and $\Lambda: \mathbb{Z}^2 \to \text{Aut}(C(X))$ is a homomorphism, then $C(X) \rtimes_{\Lambda} \mathbb{Z}^2$ can be asymptotically embedded into a unital simple AF-algebra provided that $X$ admits a strictly positive $\Lambda$-invariant probability measure. Consequently $C(X) \rtimes_{\Lambda} \mathbb{Z}^2$ is quasidiagonal if $X$ admits a strictly positive $\Lambda$-invariant Borel probability measure.