Connections between dynamical systems and crossed products of Banach algebras by $\mathbb{Z}$

Abstract

Starting with a complex commutative semi-simple regular Banach algebra $A$ and an automorphism $\sigma$ of $A$, we form the crossed product of $A$ with the integers, where the latter act on $A$ via iterations of $\sigma$. The automorphism induces a topological dynamical system on the character space $\Delta(A)$ of $A$ in a natural way. We prove an equivalence between the property that every non-zero ideal in the crossed product has non-zero intersection with the subalgebra $A$, maximal commutativity of $A$ in the crossed product, and density of the non-periodic points of the induced system on the character space. We also prove that every non-trivial ideal in the crossed product always intersects the commutant of $A$ non-trivially. Furthermore, under the assumption that $A$ is unital and such that $\Delta(A)$ consists of infinitely many points, we show equivalence between simplicity of the crossed product and minimality of the induced system, and between primeness of the crossed product and topological transitivity of the system.