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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeSep 3rd 2021

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

• 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).

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

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

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

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeSep 19th 2021

added the statement that the norm topology makes $U(\mathcal{H})$ 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.

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

added the statement that $U(\mathcal{H})_{strong}$ 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)
• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeSep 19th 2021
• (edited Sep 19th 2021)

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

I know that

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

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

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

$\mathrm{U}(\mathcal{H})_{\vert V_{\mathrm{e}}} \simeq V_{\mathrm{e}} \times S^1$

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.

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

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

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

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

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

• CommentRowNumber15.
• CommentAuthorDavidRoberts
• CommentTimeJan 15th 2022

I guess one issue is that one presumably integrates the map $S^n \to U(\mathcal{H})\hookrightarrow B(\mathcal{H})$, and then needs to know the resulting element is still inside the unitary group. Also, two such lifts to $U(\mathcal{H})$ differ by a $U(1)$-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 $L^2\big(S^n, \, \mathcal{H}\big)$ of square-integrable functions on $S^n$ with values in $\mathcal{H}$, and then use that there is presumably an isomorphism like $\mathcal{H} \;\simeq\; \mathcal{H} \otimes L^2\big( S^n,\, \mathcal{H}\big)$.

• CommentRowNumber17.
• CommentAuthorDavidRoberts
• CommentTimeJan 16th 2022

If we are thinking about measure, then of course we can delete one point from $S^n$ 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 $\mathbb{R}^n$. 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 $U(1)$, but also the Lie algebra. So perhaps thinking about the Lie algebra of $U(\mathcal{H})$ or $PU(\mathcal{H})$ (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 $\mathcal{H}$ to $\mathcal{H} \otimes L^2\big( S^n, \mathcal{H} \big)$ 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)