Banach algebra crossed products by inverse semigroup actions

TL;DR

This paper simplifies the theory of covariant representations for inverse semigroup actions on Banach algebras and provides a universal description of the associated crossed product, enabling disintegration of all representations.

Contribution

It offers a self-contained, simplified presentation and a general universal framework for Banach algebra crossed products by inverse semigroup actions.

Findings

Provides a universal disintegration theorem for representations

Simplifies the theory of covariant representations

Extends disintegration results beyond space actions

Abstract

We give a self-contained and simplified presentation of the theory of covariant representations for inverse semigroup actions on Banach algebras, which was recently introduced in the authors and A. Mckee in the twisted case. The main result of this note is a general universal description of the associated Banach algebra crossed product, that allows disintegration of all representations of the crossed product. Such a disintegration was studied so far only for actions on spaces.