The weak tracial Rokhlin property for finite group actions on simple C*-algebras

TL;DR

This paper introduces the weak tracial Rokhlin property for finite group actions on simple C*-algebras, explores its stability and examples, and extends known results about crossed products and fixed point algebras to the nonunital case.

Contribution

It develops the weak tracial Rokhlin property for nonunital simple C*-algebras, proving stability properties and extending results on crossed products and fixed point algebras.

Findings

The property is stable under restriction, tensor products, and limits.

Examples of actions with this property on stably projectionless C*-algebras.

Crossed products and fixed point algebras retain simplicity and tracial rank zero.

Abstract

We develop the concept of weak tracial Rokhlin property for finite group actions on simple (not necessarily unital) C*-algebras and study its properties systematically. In particular, we show that this property is stable under restriction to invariant hereditary C*-algebras, minimal tensor products, and direct limits of actions. Some of these results are new even in the unital case and answer open questions asked by N. C. Phillips in full generality. We present several examples of finite group actions with the weak tracial Rokhlin property on simple stably projectionless C*-algebras. We prove that if $\alpha \colon G \rightarrow \mathrm{Aut}(A)$ is an action of a finite group $G$ on a simple C*-algebra $A$ with tracial rank zero and $\alpha$ has the weak tracial Rokhlin property, then the crossed product $A \rtimes _{\alpha} G$ and the fixed point algebra $A^{\alpha}$ are simple with tracial rank zero. This extends a result of N. C. Phillips to the nonunital case. We use the machinery of Cuntz subequivalence to work in this nonunital setting.