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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2021

    am finally giving this its own entry. Nothing much here yet, though, still busy fixing some legacy cross-linking…

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 4th 2021

    Note about how the error arose in Atiyah–Segal, and that the norm topology is still distinct (and finer) on U(H).

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2021

    Thanks. BTW, it’s spelled [[norm topology]], with square brackets around it.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 4th 2021

    Sorry :-)

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2021
    • (edited Sep 4th 2021)

    Interesting to look at the time stamps:

    • In Feb 2012 Uribe et al. pick up the topic and amplify (Sec. 1) the wrong statement from Atiyah-Segal.

    • In Sept 2013 Schottenloher points out the issue.

    • In Nov 2013 Uribe et al.’s article gets published.

    • In July 2014 Uribe at al.-prime notice the issue, apparently still unaware of Schottenloher’s preprint (?).

    • In Aug 2014 Uribe et al.-prime is already published, too.

    • In 2015 nothing happens.

    • In 2016 nothing happens.

    • In 2017 nothing happens.

    • In Aug 2018 (a rewrite of) Schottenloher’s preprint is finally published.

    What gives?

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 4th 2021

    One wonders why Atiyah-Segal made the error. An early sign of what was to come?

    I tried to really in-depth read the twisted K-theory paper as a PhD student, and it was so brief on details in certain places I made no headway for a long time, and eventually gave up. A number of reasons for this, but certainly having mistakes like the one under discussion doesn’t help!

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 4th 2021
    • (edited Sep 19th 2021)

    One wonders why Atiyah-Segal made the error.

    People make mistakes all the time. Authorities do, too.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2021

    added the statement that U(ℋ)U(\mathcal{H}) in the strong topology is completely metrizable, with pointer to:

    diff, v6, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2021
    • (edited Sep 19th 2021)

    added pointer to

    for a proof that the weak and strong topology on U(ℋ)U(\mathcal{H}) agree and make it a topological group

    diff, v6, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2021

    added the statement that the norm topology makes U(ℋ)U(\mathcal{H}) a Banach Lie group.

    Schottenloher talks about this somewhat informally, while Espinoza & Uribe point to Neeb 1997. However, I don’t see that Neeb says this explicitly.

    diff, v6, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2021
    • (edited Sep 19th 2021)

    added the statement that U(ℋ) strongU(\mathcal{H})_{strong} is not locally compact, with reference to section 5 in:

    • Rostislav Grigorchuk, Pierre de la Harpe, Amenability and ergodic properties of topological groups: from Bogolyubov onwards, in: Groups, Graphs and Random Walks, Cambridge University Press 2017 (arXiv:1404.7030, doi:10.1017/9781316576571.011)

    diff, v6, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 19th 2021
    • (edited Sep 19th 2021)

    Is U(ℋ)\mathrm{U}(\mathcal{H}) well-pointed?

    I know that

    1. S 1S^1 is well-pointed;

    2. PU(ℋ)\mathrm{PU}(\mathcal{H}) is well-pointed;

    3. there is an open neighbourhood V eV_{\mathrm{e}} of e\mathrm{e} in PU(ℋ)PU(\mathcal{H}) such that

      U(ℋ) |V e≃V e×S 1\mathrm{U}(\mathcal{H})_{\vert V_{\mathrm{e}}} \simeq V_{\mathrm{e}} \times S^1

    4. products of h-cofibrations remain h-cofibrations.

    This seems like it might be getting close. Or maybe not.

    [ edit: on the other hand, it dawns on me that I don’t actually need to know the answer to do what I want to do… ]

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 20th 2021
    • (edited Sep 20th 2021)

    I have spelled out the various topologies, following the list as given by Espinoza & Uribe. Then I have further refined the list of propositions about these topologies, with references.

    It seems that all these results, except maybe concerning the compact-open topology, are already due to K-H Neeb in the 1990s.

    diff, v9, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJan 14th 2022
    • (edited Jan 14th 2022)

    Given a map f:S n⟶U(1)f \colon S^n \longrightarrow U(1), if n≥2n \geq 2 it lifts to a map f^:S n⟶ℝ 1 \hat f \colon S^n \longrightarrow \mathbb{R}^1, which can be integrated against the unit volume form of S nS^n and the result

    ∫ S nf(p)vol(p)modℤ \int_{S^n} f(p) \, vol(p) \; mod \mathbb{Z}

    is a well-defined element of U(1)U(1), depending continuously on ff and independent of the choice of lift.

    I am wondering if something like this works for ℝ 1→U(1)\mathbb{R}^1 \to U(1) replaced by U(ℋ)→PU(ℋ)U(\mathcal{H}) \to PU(\mathcal{H}) and n≥3n \geq 3?

    • CommentRowNumber15.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 16th 2022

    I guess one issue is that one presumably integrates the map S n→U(ℋ)↪B(ℋ)S^n \to U(\mathcal{H})\hookrightarrow B(\mathcal{H}), and then needs to know the resulting element is still inside the unitary group. Also, two such lifts to U(ℋ)U(\mathcal{H}) differ by a U(1)U(1)-valued function, rather than a constant integer, though I think your first observation allows us to integrate that to a constant, and this might the difference between the integrated maps.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJan 16th 2022

    I was thinking about integrating, but there does not seem to be a reason why that integral should still be unitary. Already the sum of a finite number of unitary operators is generically not unitary anymore.

    But I came to think that I should instead be passing to something like the Hilbert space L 2(S n,ℋ)L^2\big(S^n, \, \mathcal{H}\big) of square-integrable functions on S nS^n with values in ℋ\mathcal{H}, and then use that there is presumably an isomorphism like ℋ≃ℋ⊗L 2(S n,ℋ)\mathcal{H} \;\simeq\; \mathcal{H} \otimes L^2\big( S^n,\, \mathcal{H}\big).

    • CommentRowNumber17.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 16th 2022

    If we are thinking about measure, then of course we can delete one point from S nS^n to get something homeomorphic to an open disk, and then the question is whether one can find an “average” of a (bounded) family of unitary operators defined on a precompact region in ℝ n\mathbb{R}^n. For example, here’s ’a preprint considering the definition of the mean of Lie-group-valued data, including the continuous case: https://hal.inria.fr/hal-00938320

    • CommentRowNumber18.
    • CommentAuthorDavidRoberts
    • CommentTimeJan 16th 2022

    One other thing that occurs to me is that the reals are not just the 1-connected cover of U(1)U(1), but also the Lie algebra. So perhaps thinking about the Lie algebra of U(ℋ)U(\mathcal{H}) or PU(ℋ)PU(\mathcal{H}) (and then exponentiating, like the circle case) might be worth a shot. It’s the sort of thing that looks like it appears in that preprint in #17.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJan 16th 2022

    Interesting that text on averaging over Lie groups. I was wondering about that. I was also thinking about using patches onto which the exponential map is surjective, but I see no reason why the given map would be constrained to such a patch.

    The idea in #16 of expanding out ℋ\mathcal{H} to ℋ⊗L 2(S n,ℋ)\mathcal{H} \otimes L^2\big( S^n, \mathcal{H} \big) seems to do the trick for the application I have, only assuming that it works the way one would naively assume it would work.

    I may have to think more carefully about Hilbert spaces of square integrable functions with values in another (separable) infinite Hilbert space. Is there some decent textbook account on this? I see it’s used in passing here and there, e.g. from slide 17 on in Pysiak: “Representations of groupoids and imprimitvity systems” (pdf)

    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeJan 16th 2022
    • (edited Jan 16th 2022)

    Oh, I see that the keywords I need are “decomposable operators on direct integral Hilbert spaces” and a relevant monograph is

    • Jacques Dixmier, Chapter II of: Les algebres d’opérateurs dans l’espace hilbertien, Cahiers Scientifiques, fasc. 25, Gauthier-Villars,Paris, 1957

    and, for the equivariant case that I am really after:

    • Raymond C. Fabec, around IV.12 of: Fundamentals of Infinite Dimensional Representation Theory, Chapman and Hall/CRC 2019 (ISBN:9780367398408)
    • CommentRowNumber21.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 22nd 2023

    Added the result that the Banach Lie group U(ℋ) normU(\mathcal{H})_{norm} is metrizable hence paracompact, citing Nikolaus–Sachse–Wockel.

    diff, v13, current

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