Crossed products of Banach algebras. III

TL;DR

This paper extends the theory of crossed products of Banach algebras to the ordered setting, establishing a bijection between positive representations and covariant representations, and analyzing pre-ordered generalized Beurling algebras.

Contribution

It adapts the crossed product construction to pre-ordered Banach algebras and characterizes positive representations, including a uniqueness result and applications to Beurling algebras.

Findings

Established a bijection between positive representations and covariant representations.

Proved the pre-ordered crossed product is essentially unique under mild conditions.

Showed pre-ordered Beurling algebras are isomorphic to crossed products, enabling description of positive representations.

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 we adapt the theory to the ordered context. We construct a pre-ordered crossed product of a Banach algebra from a pre-ordered Banach algebra dynamical system $(A,G,\alpha)$ and a given uniformly bounded class $\mathcal{R}$ of continuous covariant representations of $(A,G,\alpha)$. If $A$ has a positive bounded approximate left identity and $\mathcal{R}$ consists of non-degenerate continuous covariant representations, we establish a bijection between the positive non-degenerate bounded representations of the pre-ordered crossed product on pre-ordered Banach spaces with closed cones and the positive non-degenerate $\mathcal{R}$-continuous covariant representations of $(A,G,\alpha)$ on such spaces. Under mild conditions, we show that this pre-ordered crossed product is the essentially unique pre-ordered Banach algebra for which such a bijection exists. Finally, we study pre-ordered generalized Beurling algebras. We show that they are bipositively topologically isomorphic to pre-ordered crossed products of Banach algebras associated with pre-ordered Banach algebra dynamical systems, and hence the general theory allows us to describe their positive representations on pre-ordered Banach spaces with closed cones.