Characterisations of crossed products by partial actions

TL;DR

This paper characterizes crossed products by partial group actions on C*-algebras using spectral subspaces, simplifying existing theorems and identifying important algebras as such crossed products.

Contribution

It generalizes and simplifies a theorem of Exel for partial automorphisms and identifies key algebras as crossed products by partial actions.

Findings

Characterization of crossed products via spectral subspaces

Simplification of Exel's theorem for partial automorphisms

Identification of Cuntz and Toeplitz algebras as crossed products

Abstract

Partial actions of discrete groups on $C^*$-algebras and the associated crossed products have been studied by Exel and McClanahan. We characterise these crossed products in terms of the spectral subspaces of the dual coaction, generalising and simplifying a theorem of Exel for single partial automorphisms. We then use this characterisation to identify the Cuntz algebras and the Toeplitz algebras of Nica as crossed products by partial actions.