nLab
Fell bundle

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Bundles

Contents

Idea

A Fell Bundle is a family of Banach spaces that varies continuously over the morphism space of a topological groupoid, where nontrivial morphisms in the groupoid induce a bilinear composition across the fibers. They are a generalization of both continuous fields of C*-algebras over topological spaces and of C *C^*-algebraic bundles (fiber bundles of C *C^*-algebras over topological groups).

Definition

Definition

A Fell Bundle over a groupoid 𝒢\mathcal{G} is a “continuous” functor E:𝒢ℭ𝔬𝔯𝔯E \colon \mathcal{G} \to\mathfrak{Corr}, where ℭ𝔬𝔯𝔯\mathfrak{Corr} is the 22-category of C *C^*-algebras and C *C^*- correspondences.

Banach algebra bundles

Mulvey considers the special case where the groupoid is a topological space and proves that such Banach space bundles are equivalent to sheaves of Banach spaces, and that both are equivalent to Banach spaces internal to the sheaf topos.

References

  • Alex Kumjian, Fell bundles over groupoids, Proceedings of the AMS, volume 126 (1998) (JSTOR)

  • Alcides Buss, Chenchang Zhu, Ralf Meyer, A higher category approach to twisted actions on C *C^*-algebras, arxiv/0908.0455

There is a variant notion of Fell bundles over inverse semigroups. Those are related to Fell bundles over the corresponding étale groupoids:

  • Alcides Buss, Ruy Exel, Fell bundles over inverse semigroups and twisted étale groupoids, J. Oper. Theory 67, No. 1, 153-205 (2012) MR2821242 Zbl 1249.46053 arxiv/0903.3388journal; Twisted actions and regular Fell bundles over inverse semigroups, arxiv/1003.0613

  • Christopher J. Mulvey, Banach sheaves, Journal of Pure and Applied Algebra 17 (1980) 69-83 doi

Last revised on April 23, 2014 at 04:28:43. See the history of this page for a list of all contributions to it.