Crossed products for actions of crossed modules on C*-algebras

TL;DR

This paper develops a framework for understanding crossed products of C*-algebras under actions of crossed modules, extending existing theories and establishing new duality and equivalence results.

Contribution

It introduces a decomposition of the crossed product functor for crossed modules, extending partial crossed product theory and duality to this setting.

Findings

Extended partial crossed products to crossed modules

Established Takesaki-Takai duality for Abelian crossed modules

Proved equivalence of categories for actions under equivalent crossed modules

Abstract

We decompose the crossed product functor for actions of crossed modules of locally compact groups on C*-algebras into more elementary constructions: taking crossed products by group actions and fibres in C*-algebras over topological spaces. For this, we extend the theory of partial crossed products from groups to crossed modules; extend Takesaki-Takai duality to Abelian crossed modules; show that equivalent crossed modules have equivalent categories of actions on C*-algebras; and show that certain crossed modules are automatically equivalent to Abelian crossed modules.