C*-crossed products by partial actions and actions of inverse semigroups

TL;DR

This paper extends the theory of partial group actions on C*-algebras to include inverse semigroup actions, showing that every partial crossed product can be viewed as a semigroup crossed product.

Contribution

It introduces a new framework for inverse semigroup actions on C*-algebras and establishes their relation to partial group actions.

Findings

Every partial crossed product is a crossed product by a semigroup action

Defines covariant representations for inverse semigroup actions

Extends the theory of partial actions to inverse semigroups

Abstract

The recently developed theory of partial actions of discrete groups on $C^*$-algebras is extended. A related concept of actions of inverse semigroups on $C^*$-algebras is defined, including covariant representations and crossed products. The main result is that every partial crossed product is a crossed product by a semigroup action.