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Is there a characterization for the compact subgroups of the unitary operators in a Hilbert space, where the unitaries are furnished with the norm topology? What about other topologies?

edited Feb 13 at 21:41

Rami
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asked Feb 13 at 20:59

Nicolas Börger
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For the strong topology, there is no characterization: any compact group $G$ embeds into the unitary group of $L^2(G)$ via the left regular representation. – Alain Valette Feb 13 at 21:21

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For the weak topology the unitaries form already a compact group. – András Bátkai Feb 13 at 21:50

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The unitary group $U(H)$ of a Hilbert space $H$, with the norm topology, has no small subgroup (its intersection with the ball of radius 1 centered at the identity operator, contains only the trivial subgroup). Therefore the same holds for a compact subgroup $G$ of $U(H)$. Isn't there a result related to Hilbert's 5th problem (by von Neumann maybe?) that implies that $G$ is a Lie group? (a not necessarily connected Lie group, of course). I've no time to look it up today... – Alain Valette Feb 13 at 22:26

According to Wikipedia, yes : "Gleason, Montgomery and Zippin characterized Lie groups amongst locally compact groups, as those having no small subgroups." – Amin Feb 13 at 22:44

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@Andras Batkai: Actually, the unitaries with the weak topology do not form a compact group. In fact, the weak and strong topologies give the same relative topology on the space of unitaries. – Jesse Peterson Feb 13 at 23:42

Narutaka OZAWA · Answer 1 · 2013-02-14 08:09:21Z UTC

In fact, for any Banach space $X$, every norm compact group $G$ of invertible operators on $X$ generates a finite-dimensional semi-simple algebra (which is isomorphic to a finite direct sum of matrix algebras). So, $G$ is indeed a Lie group as was suggested. I don't know a right reference, but such $G$ generates contractible (aka super-amenable) Banach subalgebra of $B(X)$, which has to be finite-dimensional at least when $X$ is a Hilbert space (see Paulsen--Smith, Proc. Edinb. Math. Soc. (2) 45 (2002), or my paper arXiv:1110.6216). The proof for general $X$ is bit tricky, but reduces to the Hilbertian case. (I learned it from Nicolas Monod.)

Rami · Answer 2 · 2013-02-13 22:10:01Z UTC

I think that the answer is that all such groups are Lie groups, and one can decompose the Hilbert space to a finite direct sum s.t. each summent will be a tensor product of a f.d. representation of the group and an Hilbert space with a trivial action

"proof" as Alain suggested a subgroup $G$ winch is compact w.r.t. the strong topology is the same as (faithful) unitary representation of $G$. any such representation can be decomposed to an Hilbert direct sum: $$H:=\bigoplus W_i \otimes V_i,$$ where $W_i$ are (f.d.) distinct irreducible representations of $G$ and $V_i$ are Hilbert spaces. The question is: What are the conditions on $V_i$ so that the map $G \to O(H)$ will be continuous?

I think that the answer is that all but finitely many should be $0$. This implies the above claim.

Compact subgroups of the unitary group of operators in a hilbert space

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