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• CommentRowNumber1.
• CommentAuthorTodd_Trimble
• CommentTimeJul 8th 2011

I would like to eventually improve the articles on unitary irreps of the Poincare group, but I am having some doubts that I hope one of you could clear up quickly. In the article unitary group there is no mention of the topology of $U\left(H\right)$, where $H$ is a (we’ll say separable) Hilbert space. It seems to me the most natural choice of topology would be as a subspace of $L\left(H,H\right)$, the space of bounded linear operators with the norm topology. But in reading accounts of Stone’s theorem, I am led to believe this is not the topology chosen; rather one chooses the strong operator topology, the smallest topology such that all evaluation maps ${\mathrm{ev}}_{h}:L\left(H,H\right)\to H$ are continuous.

Could someone confirm for me the correct topology on the unitary group, to be used in the article unitary group? If it is the strong operator topology, are there any nice high-level or conceptual reasons why that should be the one chosen (besides the fact that it might be needed to make Stone’s theorem rigorous)?

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJul 10th 2011

I asked my question at the Café, and John Baez kindly responded. Yes, it is apparently the so-called strong operator topology that is meant. (I didn’t ask the question about conceptual explanations.) But now I have another question: at Kuiper’s theorem it was mentioned that $U\left(H\right)$ is contractible in “the” operator topology, but when I click on that article, I found not one but several operator topologies mentioned. So which one does Kuiper’s theorem refer to?

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJul 10th 2011

Hi Todd,

probably the references that explicitly claim to be about mathematical physics do mention this. For instance The Mathematics any Physicist should know , around page 9.

Concerning Kuiper’s theorem: I guess the Wikipedia article gives the relevant pointers. Originally proven for the norm topology, the theorem does hold for the strong operator topology, too.

All the nLab entries on these matters deserve to be improved, certainly.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeJul 10th 2011
• (edited Jul 10th 2011)

Thanks, Urs – that looks like a really nice reference. I put in some changes at unitary group and Kuiper’s theorem to reflect what you have just told me.

I am still curious though as to whether a nice categorical or nPOV story could be told about the strong operator topology in particular, as this is apparently the correct topology for purposes of studying unitary representations of Lie groups.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJul 10th 2011

I am still curious though as to whether a nice categorical or nPOV story could be told about the strong operator topology in particular, as this is apparently the correct topology for purposes of studying unitary representations of Lie groups.

Yes, I think that’s a very good question. I am not sure that I know the answer, but I want to look some things up and see. I’ll try to get back to you on this. Unfortunately, right now I must be concentrating on something else.