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

TL;DR

This paper introduces a new version of the tracial Rokhlin property suitable for projection-free C*-algebras and investigates its implications for the structure of crossed product algebras under finite group actions.

Contribution

It defines the projection free tracial Rokhlin property and explores its impact on the stability and structural properties of crossed product C*-algebras.

Findings

Crossed products retain stable rank one under the new property (pending correction).

The new property applies to C*-algebras without nontrivial projections.

Provides a framework for analyzing group actions on projection-free C*-algebras.

Abstract

In this paper we introduce an analog of the tracial Rokhlin property, called the {\emph {projection free tracial Rokhlin property}}, for $C^*$-algebras which may not have any nontrivial projections. Using this we show that if $A$ is an infinite dimensional stably finite simple unital $C^*$-algebra with stable rank one, with strict comparison of positive elements, with only finitely many extreme tracial states, and with the property that every 2-quasi-trace is a trace, and if $\alpha$ is an action of a finite group $G$ with the projection free tracial Rokhlin property, then the crossed product $C^*(G, A, \alpha)$ also has stable rank one (Except there is a mistake in Lemma 3.16, so this is no longer proven)