Abstract
AbstractC*-algebras form a 2-category with *-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 the equivalence of such actions, covariant representations and saturated Fell bundles. For 2-groups, weak actions combine twists in the sense of Green, and Busby and Smith.The Packer–Raeburn Stabilization Trick implies that all Busby–Smith twisted group actions of locally compact groups are Morita equivalent to classical group actions. We generalize this to actions of strict 2-groupoids.
Highlights
An automorphism of a C∗-algebra A is called inner if it is of the form Adu : a → uau∗ for some unitary multiplier u of A
We show that weak actions of a group are equivalent to saturated Fell bundles over this group
We will generalize our definitions, allowing general 2-groupoids to act. This explains the coherence laws: they are part of the standard way to define the 3-category of 2-categories; weak group actions are the same thing as morphisms from a group, viewed as a 2-category, to another 2-category; weakly equivariant maps are transformations between such morphisms, and modifications appear under the same name in [17]
Summary
An automorphism of a C∗-algebra A is called inner if it is of the form Adu : a → uau∗ for some unitary multiplier u of A. A higher category approach to twisted actions on C∗-algebras category theory is that two functors that are related by an invertible natural transformation should be considered equivalent. This leads to the notion of equivalence of categories. There are modifications between weakly equivariant maps These generalize unitary intertwining operators between covariant representations to the setting where the target algebra carries a non-trivial action, and provide the correct notion of equivalence for weakly equivariant maps between weak actions. The same argument applies to crossed modules of locally compact topological groupoids: any topological weak action by correspondences of such a crossed module is equivalent to a strict action of the crossed module Such strictification results are rather unusual in higher category theory. Our motivating examples are the 2-categories of C∗-algebras C∗(2) and Corr(2) based on ∗-representations or correspondences, respectively, and unitary intertwiners
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
