Tracial Rokhlin property for finite group actions on non-unital simple C*-algebras

We introduce the tracial Rokhlin property for finite group actions on simple not necessarily unital C*-algebras which coincides with Phillips' definition in the unital case. We study its basic properties. Our main result is that if $\alpha:G\to \mathrm{Aut(A)}$ is an action of a finite group $G$ on a simple (not necessarily unital) C*-algebra $A$ with tracial topological rank zero and $\alpha$ has the tracial Rokhlin property, then $A \rtimes _{\alpha}G$ and $A^{\alpha}$ have tracial topological rank zero. The main idea to show this is to prove that a simple non-unital C*-algebra has tracial topological rank zero if and only if it is Morita equivalent to a simple unital C*-algebra with tracial topological rank zero. Moreover, we show that all of the following classes of (not necessarily unital) simple C*-algebras are closed under taking crossed products and fixed point algebras with actions of finite groups with the tracial Rokhlin property: simple separable C*-algebras $A$ of real rank zero with $\mathrm{TR}(A)\leq k$, simple separable C*-algebras of real rank zero, simple separable C*-algebras of stable rank one and real rank zero, simple separable nuclear $\mathcal{Z}$-stable C*-algebras, simple C*-algebras with Property~(SP), and simple separable tracially $\mathcal{Z}$-absorbing C*-algebras.