Properties of Topological Dynamical Systems and Corresponding $C^*$-Algebras

Abstract

We show the equivalence of a certain property of topological dynamical systems $\\Sigma=(X,G)$ and a particular structure of ideals in the corresponding crossed product $A(\\Sigma)=C(X)\\underset{\\alpha}{\\rtimes}G$ where $X$ is a compact set and $G$ is a discrete group.\nAs an application, we give a complete characterization for $A(\\Sigma)$ to be simple.