The Rokhlin Property for Automorphisms on Simple C*-Algebras

TL;DR

This paper investigates automorphisms with the tracial Rokhlin property on certain simple C*-algebras, showing that the crossed product retains the algebra's class under specific conditions.

Contribution

It proves that crossed products by automorphisms with the tracial Rokhlin property preserve the class of simple amenable C*-algebras satisfying the UCT and having tracial rank zero after tensoring with certain UHF algebras.

Findings

Crossed product remains in class  under specified conditions.

Automorphisms with the tracial Rokhlin property induce well-behaved crossed products.

The result extends classification results for simple C*-algebras with automorphisms.

Abstract

Let $\mathcal{A}$ be the class of unital separable simple amenable $C$*-algebras $A$ which satisfy the Universal Coefficient Theorem for which $A\otimes M_{\texttt{P}}$ has tracial rank zero for some supernatural number $\texttt{p}$ of infinite type. Let $A\in \mathcal{A}$ and let $\alpha$ be an automorphism of $A.$ Suppose that $\alpha$ has the tracial Rokhlin property. Suppose also that there is an integer $J\geq 1$ such that $[\alpha^J]=[\mbox{id}_A]$ in $KL(A,A)$, we show that $A\rtimes_{\alpha}\mathbb{Z}\in \mathcal{A}.$