A higher category approach to twisted actions on C*-algebras

TL;DR

This paper develops a higher category framework for twisted actions on C*-algebras, unifying various known concepts and extending results to 2-groupoid actions, with implications for Morita equivalence.

Contribution

It introduces a 2-category approach to twisted actions on C*-algebras, generalizing classical and groupoid cases, and extends the Packer-Raeburn Stabilisation Trick to 2-groupoids.

Findings

Identifies weak actions with known twisted actions in the group case

Shows all Busby-Smith twisted group actions are Morita equivalent to classical actions

Extends the Stabilisation Trick to strict 2-groupoid actions

Abstract

C*-algebras form a 2-category with \Star{}homomorphisms or correspondences as morphisms and unitary intertwiners as 2-morphisms. We use this structure to define weak actions of 2-categories, weakly equivariant maps between weak actions, and modifications between weakly equivariant maps. In the group case, we identify the resulting notions with known ones, including Busby-Smith twisted actions and equivalence of such actions, covariant representations, and saturated Fell bundles. For 2-groups, weak actions combine twists in the sense of Green and Busby-Smith. The Packer-Raeburn Stabilisation Trick implies that all Busby-Smith twisted group actions of locally compact groups are Morita equivalent to classical group actions. We generalise this to actions of strict 2-groupoids.