Stable and real rank for crossed products by finite groups

Abstract

A long-standing open question in the theory of group actions on C*-algebras is the stable rank of the crossed product. Specifically, N. C. Phillips asked that if a finite group $G$ acts on a simple unital C*-algebra $A$ with stable rank one, does the crossed product have stable rank one? A similar question can be asked about the real rank. Most of the existing partial answers contain a reasonable restriction (mainly, a Rokhlin-type property) on the action and assumptions on $A$. We remove all extra assumptions on $A$ (for instance, stable finiteness and that the order on projections over $A$ is determined by traces) and we prove that if the action has the tracial Rokhlin property and $A$ is simple and $\sigma$-unital with stable rank one or real rank zero, then so do the crossed product and the fixed point algebra. Moreover, we show that if the Kirchberg's central sequence algebra $\mathrm{F}(A)$ has real rank zero, then the weak tracial Rokhlin property is equivalent to the tracial Rokhlin property for actions on simple unital separable C*-algebras $A$.