  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.
