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

Part of: Categories with structure Selfadjoint operator algebras

Published online by Cambridge University Press:  30 August 2012

Alcides Buss ,
Ralf Meyer  and
Chenchang Zhu
Alcides Buss
Affiliation:
Departamento de Matemática, Universidade Federal de Santa Catarina, 88.040-900 Florianópolis-SC, Brazil (alcides@mtm.ufsc.br)
Ralf Meyer
Affiliation:
Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstraße 3–5, 37073 Göttingen, Germany (rameyer@uni-math.gwdg.de) Courant Research Centre ‘Higher Order Structures’, Georg-August-Universität Göttingen, Bunsenstraße 3–5, 37073 Göttingen, Germany (zhu@uni-math.gwdg.de)
Chenchang Zhu
Affiliation:
Courant Research Centre ‘Higher Order Structures’, Georg-August-Universität Göttingen, Bunsenstraße 3–5, 37073 Göttingen, Germany (zhu@uni-math.gwdg.de)
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Abstract

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C*-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.

Keywords

Green twisted actionBusby–Smith twisted actionFell bundle2-categoryMorita equivalencePacker–Raeburn Stabilization Trick

MSC classification

Secondary: 46L55: Noncommutative dynamical systems 18D05: Double categories, $2$-categories, bicategories and generalizations
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 56 , Issue 2 , June 2013 , pp. 387 - 426
Copyright
Copyright © Edinburgh Mathematical Society 2013

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