# Journal of Operator Theory

Volume 69, Issue 1, Winter 2013 pp. 17-31.

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

**Author institution:**Department of Mathematics Physics and Information Engineering, Jiaxing University, Jiaxing, 314001, P.R. China

**Summary:**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 $[\alpha|_{1\otimes A}]=[\mathrm{id}|_{1\otimes A}]$ in $KL(1\otimes A,C(X, A))$. We show that $C(X,A)\\ \rtimes_{\alpha} \mathbb{Z}$ has tracial rank zero.

**Keywords:**Crossed products, $\alpha$-simple, tracial rank zero

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