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Thanks. BTW, it’s spelled [[norm topology]]
, with square brackets around it.
Sorry :-)
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?
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!
One wonders why Atiyah-Segal made the error.
People make mistakes all the time. Authorities do, too.
added the statement that $U(\mathcal{H})$ in the strong topology is completely metrizable, with pointer to:
added pointer to
for a proof that the weak and strong topology on $U(\mathcal{H})$ agree and make it a topological group
added the statement that $U(\mathcal{H})_{strong}$ is not locally compact, with reference to section 5 in:
Is $\mathrm{U}(\mathcal{H})$ well-pointed?
I know that
$S^1$ is well-pointed;
$\mathrm{PU}(\mathcal{H})$ is well-pointed;
there is an open neighbourhood $V_{\mathrm{e}}$ of $\mathrm{e}$ in $PU(\mathcal{H})$ such that
$\mathrm{U}(\mathcal{H})_{\vert V_{\mathrm{e}}} \simeq V_{\mathrm{e}} \times S^1$
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… ]
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.
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 $n \geq 3$?
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.
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 $\mathcal{H} \;\simeq\; \mathcal{H} \otimes L^2\big( S^n,\, \mathcal{H}\big)$.
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
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.
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)
Oh, I see that the keywords I need are “decomposable operators on direct integral Hilbert spaces” and a relevant monograph is
and, for the equivariant case that I am really after:
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