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

We develop the concept of weak tracial Rokhlin property for finite group actions on simple (not necesserily unital) C*-algebras and study its properties systematically. In particular, we show that this property is stable under restriction to invariant hereditary C*-algebras, and minimal tensor products and direct limits of actions. Some of the results are new even in the unital case. We present several examples of finite group actions with the weak tracial Rokhlin property on simple nonunital C*-algebras, especially on 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 not necessarily unital 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 Phillips to the nonunital case. A similar result holds for simple purely infinite C*-algebras. We use the machinery of Cuntz subequivalence to deal with the lack of trace in this setting.