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

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.

References in corpus (3)

• A Categorical Approach to Imprimitivity Theorems for C*-Dynamical Systems
• Non-Hausdorff Symmetries of C*-algebras
• A bicategorical approach to Morita equivalence for rings and von Neumann algebras

Cited by in corpus (26)

• Inverse semigroup actions on groupoids
• An equivalence theorem for reduced Fell bundle C*-algebras
• Coactions and Fell bundles
• On a categorical framework for classifying C*-dynamics up to cocycle conjugacy
• Remarks on anomalous symmetries of C*-algebras
• Frobenius structures over Hilbert C*-modules
• Non-Hausdorff Symmetries of C*-algebras
• Iterated crossed products for groupoid fibrations
• Roe bimodules as morphisms of discrete metric spaces
• On bicolimits of $C^{\ast }$-categories
• Morita enveloping Fell bundles
• Colimits in the correspondence bicategory
• Zappa-Szép product of a Fell bundle and a groupoid
• Geometric foundations for classical $\mathrm{U}(1)$-gauge theory on noncommutative manifolds
• Irreducible Induced Representations of Fell Bundle C*-Algebras
• Exponential Functors, R-Matrices and Twists
• Representations of *-algebras by unbounded operators: C*-hulls, local-global principle, and induction
• Twisted K-theory with coefficients in C*-algebras
• Functor of Points and Height Functions for Noncommutative Arakelov Geometry
• Fell bundles over quasi-lattice ordered groups and ${\mathrm{C}}^{\ast }$-algebras of compactly aligned product systems
• Crossed extensions and equivalences of topological 2-groupoids
• A classical groupoid model for quantum networks
• The Imprimitivity Fell Bundle
• Crossed module actions on continuous trace $C^{\ast }$-algebras
• The bicategory of topological correspondences
• The bicategory of groupoid correspondences