I have

added more original references (such as Illusie and Dixmier & Duady);

added hyperlinks to the actual definitions of all the topologies involved;

re-organized the statement of the proof to first talk about $U(\mathcal{H})$ (as all the references do) and only mention $GL(\mathcal{H})$ afterwards (instead of the other way around)

In the process of adding comments in the references-section on who proved which aspect, I incrementally ended up replacing essentially all lines the entry used to have on this (some of which I found were confusing in their terminology). So now I have ended up with essentially a complete rewrite of the entry (short as it may be), even though this wasn’t my intention, originally. If you want to double-check that I didn’t erase anything that should be kept, compare to rev 5.

(The line about which topologies make $U(\mathcal{H})$ a CW complex or ANR or not should better be discussed at U(ℋ), not here.)

]]>I’ve updated Kuiper’s theorem slightly to point out that in fact most of the various topologies on $U(H)$ weaker than the norm topology agree. This is due to Espinoza-Uribe, which an earlier partial contribution by Schottenloher. So it seems that Andrew’s complaint in #6 was probably directed at $B(H)$ or $GL(H)$, since the weak operator and strong operator topologies agree on $U(H)$, but not on $GL(H)$.

I’ve also edited unitary group to add this reference, and I found that the page claimed that $U(H)$ was the maximal compact subgroup of $GL(H)$ *even in the infinite-dimensional setting* (!). So I definitely fixed that.

A note to myself to add later: $U(H)$ is a Banach Lie group in the norm topology, but not a Lie group in the strong topology; conversely, the left regular representation of a compact topological group with Haar measure is not continuous in the norm topology, but is continuous in the strong topology.

]]>U(2n) which rotates the first

n directions to the last

n directions and which therefore maps

(A,I n) to

(I n,A), hence

U(oo) is homotopy commutative, the first step on the way to being an infinite loop space ]]>

Thanks, Andrew. Somebody should add that to the entry.

About the center: that’s what I was thinking, too, but I thought I must be missing something. Maybe not.

]]>The original paper proved that it was contractible in the operator topology. Atiyah and Segal note in their paper on twisted K-theory that there is an easy proof of contractibility in the weak topology. One major difference in the topologies is that with the operator topology then it is a CW complex but with the weak topology then it isn’t even an ANR (something that annoys me intensely).

I would also define $U(\infty)$ as $\lim U(n)$ and I believe that its centre is trivial: any $A \in U(\infty)$ must be in some $U(n)$, but then there is an element in $U(2n)$ which rotates the first $n$ directions to the last $n$ directions and which therefore maps $(A,I_n)$ to $(I_n,A)$, hence if $A$ is in the centre it must be the identity.

]]>You need to specify the topology to say it is contractible. The strength of the theorem is that it is contractible in several topologies, none of which I can recall at the moment.

And I would define U(oo) as lim U(n), but the inclusion maps don’t preserve the centre, so I’m not sure about Urs’ question.

]]>I have no idea. Mike, Todd ?

]]>while we are at it: what is the center of $U(\infty) = \Omega {\lim_\to}_n B U(n)$?

]]>Looks good – being precise :)

]]>following a suggestion by Zoran, I have created a stub (nothing more) for Kuiper’s theorem

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