The structure of crossed products by endomorphisms

We describe simplicity of the Stacey crossed product A\times_\beta \N in terms of conditions of the endomorphism \beta. Then, we use a characterization of the graph C*-algebras C*(E) as the Stacey crossed product C*(E)^\gamma\times_{\beta_E}\N to study its ideal properties, in terms of the (non-classical) C*-dynamical system (C*(E)^\gamma, \beta_E). Finally, we give sufficient conditions for the Stacey crossed product A\times_\beta \N being a purely infinite simple C*-algebra.