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

TL;DR

This paper proves that crossed products of certain *-algebras by automorphisms with specific properties retain tracial rank zero, extending classification results for these dynamical systems.

Contribution

It establishes that under specified conditions, the crossed product of a *-algebra with an automorphism has tracial rank zero, advancing understanding of their structure.

Findings

Crossed products have tracial rank zero under given conditions.

Automorphisms with trivial $KL$-class preserve tracial rank.

Results apply to *-algebras with tracial rank zero and UCT.

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.