  Let $A$ be a stably finite simple unital $C^*$-algebra and suppose $\alpha $
is an action of a finite group $G$ with the tracial Rokhlin property. Suppose
further $A$ has real rank zero and the order on projections over $A$ is
determined by traces. Then the crossed product $C^*$-algebra $C^*(G,A, \alpha)$
also has real rank zero and order on projections over $A$ is determined by
traces. Moreover, if $A$ also has stable rank one, then $C^*(G,A, \alpha)$ also
has stable rank one.
