Dynamical systems associated with crossed products

TL;DR

This paper explores the relationship between algebraic and Banach algebra crossed products of commutative algebras with integers and the underlying dynamical systems, revealing how algebraic properties reflect topological dynamics.

Contribution

It introduces a Banach algebra crossed product framework and connects algebraic properties of these crossed products with the dynamics of associated systems, extending previous results.

Findings

Characterization of ideal structures via dynamical systems

Simplified proofs of existing algebraic crossed product results

New insights into ideal intersection properties in crossed products

Abstract

In this paper, we consider both algebraic crossed products of commutative complex algebras A with the integers under an automorphism of A, and Banach algebra crossed products of commutative C^*-algebras A with the integers under an automorphism of A. We investigate, in particular, connections between algebraic properties of these crossed products and topological properties of naturally associated dynamical systems. For example, we draw conclusions about the ideal structure of the crossed product by investigating the dynamics of such a system. To begin with, we recall results in this direction in the context of an algebraic crossed product and give simplified proofs of generalizations of some of these results. We also investigate new questions, for example about ideal intersection properties of algebras properly between the coefficient algebra A and its commutant A'. Furthermore, we introduce a Banach algebra crossed product and study the relation between the structure of this algebra and the topological dynamics of a naturally associated system.