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nLab > Latest Changes: Fell bundles

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  1.  
    • CommentRowNumber1.
    • CommentAuthordavidoslive
    • CommentTimeFeb 26th 2014
    I've created a page for Fell Bundles. It's only really a stub at the moment but I'll get around to expanding it eventually. The nLab POV of Fell bundles looks very different from the classical view but the two views can easily be reconciled (which I guess should form part of the expansion).
    • CommentRowNumber2.
    • CommentAuthordavidoslive
    • CommentTimeFeb 26th 2014
    I should add, it's linked from the Continuous Fields of C*-algebras page at the moment.
    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 26th 2014

    Thanks!

    I have added a few more hyperlinks and added the plural redirects. Also added a reference (but maybe not the most canonical one, more should be added)

    • CommentRowNumber4.
    • CommentAuthorspitters
    • CommentTimeFeb 26th 2014

    There is a way to treat these as a sheaf of Banach algebras and here. I would need to look a bit for the correct references. We have developed a bit of constructive Banach algebra theory that should be applicable to such bundles.

    • CommentRowNumber5.
    • CommentAuthordavidoslive
    • CommentTimeFeb 27th 2014
    Nice one - eventually this entry should definitely consider both perspectives, given the duality of sheaves and bundles. I have tried looking at Fell bundles as sheaves before but for some reason it never seemed quite right. A great excuse to revisit them, I guess.
    • CommentRowNumber6.
    • CommentAuthorElkai
    • CommentTimeMar 3rd 2014
    • (edited Mar 3rd 2014)
    Hi,
    I think the notion of C*-correspondence between C*-algebras (weakening that of C*-homomorphism) gives the right definition of Fell bundles in terms of functor ; but then the functor goes from the groupoid to the (2-)category of C*-algebras (morphisms being C*-correspondences). I am adding a page on this 2-category.
    • CommentRowNumber7.
    • CommentAuthorElkai
    • CommentTimeMar 3rd 2014
    I made a few changes in the definition.
    • CommentRowNumber8.
    • CommentAuthorspitters
    • CommentTimeMar 3rd 2014

    Is there a more categorical description/construction of Hilbert-modules and of C*-correspondences? Why is this the “right” definition?

    • CommentRowNumber9.
    • CommentAuthorElkai
    • CommentTimeMar 4th 2014
    • (edited Mar 4th 2014)

    The 2-category CorrCorr has C *C^*-algebras as objects, C *C^*-correspondences as 1-morphisms, and intertwining unitaries as 2-morphisms. 1-isomorphisms in CorrCorr are exactly Morita equivalences of C *C^*-algebras. Now, a saturated Fell bundle (A g) g∈G(A_g)_{g\in G} (cf. Kumjian’s definition) over a group GG, say, is nothing but an action of GG by Morita equivalences on the C *C^*-algebra A eA_e (every fibre A gA_g is an isomorphism A e→A eA_e \rightarrow A_e in the category Corr). More generally, a non-necessarily saturated Fell bundle over a group is just an action by correspondences on the fibre over the neutral element.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMar 4th 2014
    • (edited Mar 4th 2014)

    Thanks for all this.

    This “C*-correspondence” is modeled on the concept of _Hilbert bimodule_ (generalizing it from left bounded action to more general action, I suppose, there should be a comment on that).

    • CommentRowNumber11.
    • CommentAuthorspitters
    • CommentTimeApr 9th 2014

    Added a reference to the paper by Mulvey.

    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeApr 10th 2014

    11: Thanks. I added the doi link to the reference and completed the author’s name to the form in the original publication.

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