Let \(A\) be a unital \(AH\)-algebra and let \(\alpha \in \text{Aut}(A)\) be an automorphism. The paper is devoted to the problem when \(A\rtimes_{\alpha} \mathbb{Z}\) can be embedded into a unital simple \(AF\)-algebra. A necessary condition is that \(A\) admits a faithful \(\alpha\)-invariant tracial state. The author gives a sufficient condition for \(A\rtimes_{\alpha} \mathbb{Z}\) to be imbedded into a unital simple \(AF\)-algebra. It is proved that, when there exists an automorphism \(\kappa\) with \(\kappa_{*1}= - \text{id}_{K_1(A)}\) such that \(\alpha\circ \kappa\) and \(\kappa \circ \alpha\) are asymptotically unitarily equivalent, 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.
The author also shows that if \(A\) is a unital \(A\mathbb{T}\)-algebra, then \(A\rtimes_{\alpha} \mathbb{Z}\) can be embedded into a unital simple algebra if and only if \(A\) admits a faithful \(\alpha\)-invariant tracial state.
If \(X\) is a compact metric space and \(\Lambda: \mathbb{Z}^2\rightarrow \text{Aut}(C(X))\) is a homomorphism, then \(C(X)\rtimes_{\alpha} \mathbb{Z}^2\) can be asymptotically embedded into a unital simple \(AF\)-algebra provided that \(X\) admits a strictly positive \(\Lambda\)-invariant probability measure.
Consequently, \(C(X)\rtimes_{\alpha} \mathbb{Z}^2\) is quasidiagonal if \(X\) admits a strictly positive \(\Lambda \)-invariant Borel probability measure.