Hulls and kernels from topological dynamical systems and their applications to homeomorphism $C^{*}$ -algebras

Abstract

For a topological dynamical system $\Sigma$$=(X,\sigma )$ where $\sigma$ is a homeomorphism in an arbitrary compact Hausdorff space $X$, we consider the noncommutative hulls and kernels with respect to the action $\sigma$ in the associated $C\text{*}-$ algebra $A\left(\Sigma \right)$. We show that several ideals important for the structure of $A\left(\Sigma \right)$ have the form of such kernels and give topological characterizations of their hulls from the behavior of orbits in the dynamical system.