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Banach algebra crossed products by inverse semigroup actions

K. Bardadyn, B. K. Kwaśniewski

TL;DR

The paper develops a universal framework for Banach algebra crossed products by inverse semigroup actions, extending group-action theory to noncommutative Banach algebras. It introduces a robust notion of covariant representations, leverages double duals and Arens products to realize disintegration of representations, and defines the crossed product $A\rtimes_α S$ as the Hausdorff completion of $\ell^1(α)$ with respect to covariant representations. The main result is a bijection between representations of $A\rtimes_α S$ and covariant representations of the action, providing a disintegration mechanism independent of groupoid models. The framework unifies and extends existing $C^*$-algebra crossed product constructions, clarifying normalization and dual-predual aspects and ensuring compatibility with classical constructions in the untwisted and twisted settings.

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.

Banach algebra crossed products by inverse semigroup actions

TL;DR

The paper develops a universal framework for Banach algebra crossed products by inverse semigroup actions, extending group-action theory to noncommutative Banach algebras. It introduces a robust notion of covariant representations, leverages double duals and Arens products to realize disintegration of representations, and defines the crossed product as the Hausdorff completion of with respect to covariant representations. The main result is a bijection between representations of and covariant representations of the action, providing a disintegration mechanism independent of groupoid models. The framework unifies and extends existing -algebra crossed product constructions, clarifying normalization and dual-predual aspects and ensuring compatibility with classical constructions in the untwisted and twisted settings.

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.
Paper Structure (4 sections, 15 theorems, 39 equations)

This paper contains 4 sections, 15 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Prologue: group actions and their crossed products
  3. Covariant representations of inverse semigroup actions
  4. Crossed products for inverse semigroup actions

Key Result

Proposition 1

Let $\alpha: G\curvearrowright A$ be a group action on a Banach algebra $A$. For any covariant representation $(\pi,v)$ of $\alpha$ the formula where the series is norm convergent, defines a representation $\pi\rtimes v$ of $A\rtimes_\alpha G$, and every non-degenerate representation of $A\rtimes_\alpha G$ is of this form.

Theorems & Definitions (43)

  • Proposition 1
  • Theorem 2
  • Example
  • Definition 3
  • Remark 4
  • Remark 5
  • Example
  • Example
  • Definition 6
  • Remark 7
  • ...and 33 more