The Tracial Rokhlin Property for Automorphisms on Non-Simple C*-algebras

TL;DR

This paper investigates conditions under which crossed product C*-algebras formed by automorphisms with Rokhlin properties on unital AF-algebras have tracial rank zero, extending understanding of their structural properties.

Contribution

It establishes that if an automorphism with Rokhlin property on an -algebra is -simple and satisfies a specific K-theoretic condition, then the crossed product has tracial rank zero.

Findings

Crossed product -algebras have tracial rank zero under given conditions.

Automorphisms with Rokhlin property induce favorable structural properties.

K-theoretic conditions ensure the desired rank property in the crossed product.

Abstract

Let A be a unital AF-algebra (simple or non-simple) and let \alpha be an automorphism of A. Suppose that \alpha has certain Rokhlin property and A is \alpha-simple. Suppose also that there is an integer J\geq1 such that \alpha^{J}_{*0}=id_{K_{0}(A)}, we show that A\rtimes_{\alpha}\mathbb{Z} has tracial rank zero.