Dynamical Systems Associated with Crossed Products

Abstract

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.

On a correspondence between ideals and commutativity in algebraic crossed products 1

“…We may conclude that Theorem 2.2 is a generalization of certain parts of Corollary 4.5 in [6] and Theorem 4.5, Theorem 4.6, Corollary 4.7, Theorem 6.2 in [7].…”

“…In [5,6,7] the authors studies crossed product algebras associated to dynamical systems. Suppose that we are given a nonempty set X and a bijection σ : X → X.…”

Commutativity and Ideals in Pre-crystalline Graded Rings

“…It has been shown in [22,23,[27][28][29][30][31][32][33], that for some types of algebraic crossed products as well as C * -crossed products, there is a connection between these two assertions. Under some conditions on the crossed products the two statements are in fact equivalent, but not in general.…”

On the Exel Crossed Product of Topological Covering Maps

“…The property of topological freeness has also been observed to be equivalent or closely linked to the position of the algebra of continuous functions inside the crossed product, namely with whether it is a maximal abelian subalgebra or not. (For recent developments in this direction for reversible dynamical systems see also [43][44][45][46].) This interplay has been considered both for the universal crossed product C * -algebra and for the reduced crossed product C * -algebra, the later providing one of the important insights into the significance of those properties for representations of the crossed product.…”