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New entry Banach bundle covering for now also more special notion of Hilbert bundle and a different notion of Banach algebraic bundle. Sanity check is welcome!
Added a stub for Hilbert module and made some small changes to involution (should we distinguish antiinvolutions, unlike most people in *-algebras ?) and inner product space (quaternions etc. should be taken care of, thus carefully with left vs. right linearity).
I have a sanity-check question (on the page) about Hilbertisable Banach bundles.
I have one on inner product too.
Regarding antiinvolutions, I don’t know about the terminology, but we need antihomomorphism, so I wrote it.
I agree, from categorical point of view there is no difference between a Hilbertisable Banach bundle and Hilbert bundle, it is a property. Traditionally, even for Banach spaces, one distinuishes between a Banach space and underlying Hilbert space of the Banach space with a norm satisfying the parellelogram identity. I am just a traditional guy, feel free phrasing it in a more modern way here if you feel appropriate.
Added some references to Banach algebra.
I would have expected some sort of local triviality condition, or is this not the right sort of bundle?
Right, in this context it is for most purposes not the sensible condition.
Ah, I see. Now that I’ve clicked on bundle then I see that my default assumption “bundle = locally trivial fibre bundle” does not apply.
Do you think that locally trivial Banach fibre bundles should be on a separate page? There’s a few interesting things to do with continuity of transition functions that are worth recording. I guess it’s sufficiently different to what is on Banach bundle that a separate page seems the better plan.
Which significant reference discusses locally trivial Banach fibre bundles ? For fiber bundles with typical fiber $F$ one can restrict the group of automorphisms to some fixed subgroup $Coor F$ of the automorphism group; e.g. bounded linear maps or whatever. Then the classification is just the classification of the principal bundles with this group. So I do not see much specifics from the fiber bundles with prescribed $G = Coor F$ (Postnikov says $(G,F)$-bundles). The subtleties of continuities etc. are however finer in the case of general Banach bundles in the standard sense (on which there is huge literature and lots of motivation in operator algebras).
That’s really my point. In differential topology, we use these things all the time. It’s clear that this page, Banach bundle, is aimed at something else. So stuff about Banach bundles with regard to differential topology (aka locally trivial Banach fibre bundles) should be on a separate page.
Where Banach bundles appear in differential topology ? I know that in late 1960s it was popular to generalize the theorems on differentiable manifolds to those modelled on Banach spaces, but applications shown that those are rare and that the Frechet manifolds and manifolds based on convenient spaces are more natural in applications. What are you referring to ?
I had in mind things like twisted K-theory and my variations on the theme of “orthogonal structure”. Whilst one could argue that these aren’t quite Banach bundles, anyone learning about them ought to know the basics of Banach bundles beforehand.
Back in the 1970s when I was in Cork, my colleague Tony Seda published several articles on Banach Bundles and we jointly ran a seminar on them. The important names, including Fell, were Dauns and Hofmann and their work was linked with representation theory of C*-algebras. In that general context no local triviality condition is sensible … but I forget the details.
Good Andrew, there is a really good motivation to have it separately then, please go on with a new entry and crosslink :)
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