Crossed products by $\alpha$-simple automorphisms on $C^*$-algebras $C(X,A)$

Abstract

Let $X$ be a Cantor set, and let $A$ be a unital separable simple amenable $C$*-algebra with tracial rank zero which satisfies the Universal Coefficient Theorem, we use $C(X,A)$ to denote the set of all continuous functions from $X$ to $A$, let $\alpha$ be an automorphism on $C(X,A)$. Suppose that $C(X,A)$ is $\alpha$-simple and $[\alpha]=[\mbox{id}_{1\otimes A}]$ in $KL(1\otimes A,1\otimes A)$, we show that $C(X,A)\rtimes_{\alpha}\mathbb{Z}$ has tracial rank zero.