nLab Fell bundle

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 $C^*$-algebraic bundles (fiber bundles of $C^*$-algebras over topological groups).

Definition

Definition

A Fell Bundle over a groupoid $\mathcal{G}$ is a “continuous” functor $E \colon \mathcal{G} \to\mathfrak{Corr}$, where $\mathfrak{Corr}$ is the $2$-category of $C^*$-algebras and $C^*$- 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 $C^*$-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