Crossed product C*-algebras by finite group actions with the tracial Rokhlin property

TL;DR

This paper shows that for certain finite group actions with the tracial Rokhlin property on specific C*-algebras, key structural properties like real rank zero, stable rank one, and trace-determined projections are preserved in the crossed product.

Contribution

It establishes that under the tracial Rokhlin property, the crossed product retains important regularity properties of the original algebra, extending previous results to a broader class of actions.

Findings

Crossed product preserves real rank zero.

Order on projections remains trace-determined.

Stable rank one is preserved if initially present.

Abstract

Let $A$ be a stably finite simple unital $C^*$-algebra and suppose $\alpha $ is an action of a finite group $G$ with the tracial Rokhlin property. Suppose further $A$ has real rank zero and the order on projections over $A$ is determined by traces. Then the crossed product $C^*$-algebra $C^*(G,A, \alpha)$ also has real rank zero and order on projections over $A$ is determined by traces. Moreover, if $A$ also has stable rank one, then $C^*(G,A, \alpha)$ also has stable rank one.