Abstract
For a given $C^{*}$-dynamical system (A, G, .alpha.) with a G-simple $C^{*}$-algebra A (that is A has no proper .alpha.-invariant ideal) many authors have studied the simplicity of a $C^{*}$-crossed product A $x_{\alpha{r}}$ G. In [1] topological freeness of an action is shown to guarantee the simplicity of the reduced $C^{*}$-crossed product A $x_{\alpha{r}}$ G when A is G-simple. In this paper we investigate the pure infiniteness of a simple $C^{*}$-crossed product A $x_{\alpha}$ G of a purely infinite simple $C^{*}$-algebra A and a topologically free action .alpha. of a finite group G, and find a sufficient condition in terms of the action on the spectrum of the multiplier algebra M(A) of A. Showing this we also prove that some extension of a topologically free action is still topologically free.