Interesting examples of non-locally compact topological groups

Harmonic analysis is concentrated mostly on studying locally compact groups. I would like to ask people about examples of non-locally compact topological groups that are interesting in connection with some applications in mathematics (or physics). I must confess that I know only two examples:

• topological vector spaces are studied as examples of topological groups (with the additive group operation) to which Pontryagin duality is sometimes transferred from the class of locally compact abelian groups,

• the groups of diffeomorphisms of smooth manifolds are studied in the theory of infinite dimensional manifolds.

Can people enlighten me about other similar subjects? (If possible, with motivations.)

• Sergei, i think it might be a good idea (for the visibility of the question) to add the mathematical physics tag here. – Konstantinos Kanakoglou Apr 14 '19 at 14:48
• Konstantinos, thank you, I did it! – Sergei Akbarov Apr 14 '19 at 14:49
• I don't think that harmonic analysis studies locally compact groups, but rather that locally compact groups is a natural setting for harmonic analysis. Also many people study locally compact groups and are not involved in harmonic analysis. – YCor Apr 14 '19 at 16:46
• A huge well-studied class of topological groups is Banach spaces. Since continuous group homomorphisms between Banach spaces are bounded operators, it can really be considered as a subclass. Another (not disjoint) class is that of automorphism group of relational structures, that is, closed subgroups of symmetric groups. The very first example is the group of permutations of an infinite countable set (whose Polish group topology was introduced by L. Onofri in 1927). – YCor Apr 14 '19 at 16:50
• @SergeiAkbarov Presumably the standard one, induced by inclusion in $\mathbb R$ – Wojowu Apr 15 '19 at 6:17

• Francois, could you, please, say a few words on where this is used? – Sergei Akbarov Apr 15 '19 at 5:09
• @SergeiAkbarov As with all groups, one studies their homogeneous symplectic manifolds, resp. (projective) unitary representations as potential models of classical, resp. quantum systems. Pressley-Segal (1986, introduction) cite some such successes, but beyond that I see mainly hope and l’art pour l’art fun... – Francois Ziegler Apr 15 '19 at 6:05

There are two quite popular areas related to "big" or "large" groups. One concerns extreme amenability and fixed point properties (see Pestov), and the other concerns harmonic analysis and the representation theory (see Borodin - Olshanski).

• R W as far as I can see, they change the definition of topological group. In their considerations the operation of multiplication should not be jointly continuous, but separately continuous, is it? – Sergei Akbarov Apr 14 '19 at 17:08
• @Sergei Akbarov - Where do you see this? – R W Apr 14 '19 at 17:32
• @Sergei Akhbarov The multiplication in the unitary group is jointly continuous in the operator topology. – user95282 Apr 14 '19 at 18:17
• @Sergei Akhbarov Not so much representations of $U(\ell^2)$, but group representations in $U(\ell^2)$. See G.W. Mackey, Bull AMS 3,1 (1980) 543-698. – user95282 Apr 14 '19 at 20:35
• @Sergei Akbarov: See my comment to Peter Michor's answer. – André Henriques Apr 16 '19 at 21:42

A couple of common classes of examples you may have overlooked:

• The rationals $$\mathbb{Q}$$ with their usual topology. More generally, any non-closed subgroup of a locally compact Hausdorff group will be a non-locally compact group (in its subspace topology). There are lots of situations where it's convenient to prove something about a locally compact group by working on a dense subgroup.

• Infinite products of non-compact groups with their product topology. Some of these are topological vector spaces like $$\mathbb{R}^\omega$$, but for instance, $$\mathbb{Z}^\omega$$ is another simple example (it's homeomorphic to Baire space, and to the set of irrationals in $$\mathbb{R}$$). Products arise naturally any time you want to say "give me a whole sequence of these". The failure of such groups to have a Haar measure is a frequent sticking point in analysis and probability.

• Nate, to tell the truth, I don't understand how can separating $\mathbb Q$ from $\mathbb R$ be useful if the topology is preserved. – Sergei Akbarov Apr 15 '19 at 19:30
• @SergeiAkbarov, perhaps you might ask a number theorist …? – LSpice Apr 15 '19 at 19:50

Quantum mechanics: The unitary group $$U$$ of a Hilbert space. If you use the norm topology, then it is a Banach Lie group with bounded skew hermitian operators as Lie algebra. But unbounded self-adjoint operators $$A$$ (Schrödinger operators) lead to 1-parameter semigroups $$e^{itA}$$ in $$U$$, where now the strong operator topology plays a role.

• Some relevant facts about $U(H)$ in its strong operator topology (SOT) can be found here: arxiv.org/abs/1309.5891. In particular it is a topological group (as explained in the linked paper, a number of prominent references explicitly state that $(U(H),$SOT$)$ is not a topological group!). More precisely, it is a Polish group. Even more, when restricted to $U(H)\subset B(H)$, the strong operator topology, the weak operator topology, the ultra-strong topology, the ultra-weak topology, and the compact open topology all agree. – André Henriques Apr 16 '19 at 5:48

This is far from the use of topological groups in analysis, but:

In homotopy theory a basic background fact that is that a (based connected) space is always the classifying space of a topological group. In a little more detail: the following four categories (with suitable notions of weak homotopy equivalence) are equivalent for purposes of homotopy theory.

1. Based path-connected spaces
2. Based connected simplicial sets
3. Simplicial groups
4. Topological groups

So in some sense arbitrary topology groups play a role in homotopy theory.

In practice homotopy-theorists work with 2 as a substitute for 1, and with 3 as a tool for 2, but 4 is part of the picture. Roughly speaking, the topological group associated with a based space $$X$$ is the space of based loops in $$X$$, but in order to represent this homotopy type by an actual topological group one ordinarily goes through simplicial groups.

The question mentions topological vector spaces, but aren't Banach spaces a special case of sufficient interest to merit explicit mention? Every infinite dimensional Banach space is a non-locally compact topological group under addition and the norm topology.