Stable and real rank for crossed products by finite groups

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 \(A\).