[Paper Review] C*-crossed products by partial actions and actions of inverse semigroups

Nándor Sieben|ArXiv.org|Feb 21, 1996

Advanced Operator Algebra Research5 references6 citations

TL;DR

This paper extends the theory of partial actions of discrete groups on C*-algebras by introducing actions of inverse semigroups on C*-algebras, defining covariant representations and crossed products. The key result establishes that every partial crossed product is isomorphic to a crossed product by an action of an inverse semigroup, thereby unifying and generalizing the framework of partial actions within the broader context of inverse semigroup actions.

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.

Motivation & Objective

• To extend the theory of partial actions of discrete groups on C*-algebras beyond group actions to include actions of inverse semigroups.
• To define covariant representations and crossed products for actions of inverse semigroups on C*-algebras.
• To establish a structural connection between partial crossed products and crossed products by inverse semigroup actions.
• To provide a categorical and algebraic framework that unifies partial actions and inverse semigroup actions in C*-dynamical systems.
• To generalize existing results on partial crossed products by embedding them into a broader class of semigroup actions.

Proposed method

• Introduces the concept of an action of an inverse semigroup on a C*-algebra, generalizing group actions.
• Defines covariant representations of inverse semigroup actions on C*-algebras using a compatibility condition between the action and the representation.
• Constructs the crossed product C*-algebra as the universal C*-algebra generated by the action and the representation.
• Uses the structure of inverse semigroups to model partial symmetries and partial isomorphisms in C*-algebras.
• Applies the theory of partial actions to show that every partial crossed product arises as a crossed product by an inverse semigroup action.
• Employs universal properties and C*-algebraic techniques to prove the isomorphism between partial crossed products and inverse semigroup crossed products.

Experimental results

Research questions

• RQ1 Can partial actions of discrete groups on C*-algebras be embedded within a broader framework of actions by inverse semigroups?
• RQ2 What are the necessary and sufficient conditions for a C*-algebra to admit a covariant representation under an inverse semigroup action?
• RQ3 Is every partial crossed product isomorphic to a crossed product arising from an action of an inverse semigroup?
• RQ4 How do the algebraic and topological properties of inverse semigroups relate to the structure of the resulting crossed products?
• RQ5 What is the universal property of the crossed product associated with an inverse semigroup action?

Key findings

• Every partial crossed product is isomorphic to a crossed product arising from an action of an inverse semigroup on a C*-algebra.
• The construction of crossed products via inverse semigroup actions generalizes and unifies the theory of partial actions.
• Covariant representations for inverse semigroup actions are defined and shown to generate the universal crossed product C*-algebra.
• The paper establishes a functorial correspondence between partial actions and inverse semigroup actions, preserving the crossed product structure.
• The universal property of the crossed product ensures that any covariant representation factors through the canonical quotient map.
• The result provides a new algebraic framework for studying noncommutative dynamics and Fell bundles via inverse semigroups.