Crossed products of Banach algebras. II

TL;DR

This paper extends the theory of crossed products of Banach algebras, explores their dependence on representation classes, studies generalized Beurling algebras, and connects these concepts to classical results and module structures.

Contribution

It develops the theory further by analyzing the dependence on representation classes, introduces generalized Beurling algebras, and relates anti-representations to covariant pairs within Banach algebra dynamical systems.

Findings

Crossed product dependence on representation class analyzed.

Generalized Beurling algebras shown to be isomorphic to crossed products.

Classical Beurling algebra results recovered as special cases.

Abstract

In earlier work a crossed product of a Banach algebra was constructed from a Banach algebra dynamical system $(A,G,\alpha)$ and a class $\mathcal{R}$ of continuous covariant representations, and its representations were determined. In this paper the theory is developed further. We consider the dependence of the crossed product on the class $\mathcal{R}$ and its essential uniqueness. Next we study generalized Beurling algebras: weighted Bochner spaces of $A$-valued functions on $G$ with a continuous multiplication. Though not Banach algebras in general, they are isomorphic to a crossed product of a Banach algebra, and the earlier work therefore predicts the structure of their representations. Classical results for the usual Beurling (Banach) algebras of scalar valued functions are then retrieved as special cases. We also show how, e.g., an anti-covariant pair of anti-representations of $A$ and $G$ can be viewed as a covariant pair for a related Banach algebra dynamical system, so that the earlier work becomes applicable to classes of such other pairs. After including material on the representations of the projective tensor product of Banach algebras, we combine this idea with the results already obtained and describe the two-sided modules over the generalized Beurling algebras, where again specializing to the scalars gives a classical result.