The Unitary Group in Its Strong Topology

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1. Introduction

The unitary group plays an essential role in many areas of mathematics and physics, e.g. in representation theory, number theory, topology and in quantum mechanics. In some of the corresponding research articles complicated proofs and constructions have been introduced in order to circumvent the assumed fact that the unitary group is not a topological group when equipped with the strong topology (see Remark 1 below for details). However, in Proposition 1 it is proven that is indeed a topological group with respect to the strong topology. Moreover, in this paper it is shown that the compact open topology and the strong topology agree on, and that this topology is metrizable and contractible if is separable (and infinite dimensional). To demonstrate the relevance of these topological considerations it is shown that these results lead to a straightforward classifications of Hilbert bundles. Furthermore, the possibility of finding a Lie structure on with respect to the strong topology is discussed in a new line the following header.

2. The Unitary Group as a Topological Group

It is easy to show and well-known that the unitary group―the group of all unitary operators on a complex Hilbert space―is a topological group with respect to the norm topology on. However, for many purposes in mathematics the norm topology is too strong. For example, for a compact topological group G with Haar measure μ the left regular representation on

(1)

is continuous for the strong topology on, but L is not continuous when is equipped with the norm topology, except for finite G. This fact makes the norm topology on useless in representation theory and its applications as well as in many areas of physics or topology. The continuity property which is mostly used in case of a topological space W and a general Hilbert space and which seems to be more natural is the continuity of a left action of W on

(2)

in particular, in case of a left action of a topological group G on: Note that the above left regular representation is continuous as a map:.

Whenever is a unitary action (i.e. is a unitary operator for all) the continuity of is equivalent to the continuity of the induced map

(3)

with respect to the strong topology on. In fact, if the action is continuous then is strongly continuous by definition of the strong topology. The converse holds since is a uniformly bounded set of operators. The corresponding statement for the general linear group of bounded invertible operators holds for the compact open topology on instead of the strong topology. On the two topologies coincide, see Proposition 2 below.

We come back to the continuity of unitary actions in a broader context at the end of this paper where we elucidate the significance of the fact that is a topological group for the classification of Hilbert bundles over paracompact spaces X.

Proposition 1: is a topological group with respect to the strong topology.

Proof. Indeed, the composition is continuous: Given let be a neighbourhood of of the form where and. Now,

is a neighbourhood of and for we have:

(4)

(5)

i.e.. To show that is continuous in let a typical neighbourhood of in. For let satisfy. Then

(6)

i.e.. +

Remark 1. This result with its simple proof is only worthwhile to publish because in the literature at several places the contrary is stated and because therefore some extra but superfluous efforts have been made. For example, Simms [1] explicitly states that the unitary group is not a topological group in its strong topology and that therefore the proof of Bargmann’s theorem [2] has to be rather involved. But also recently in the paper of Atiyah and Segal [3] some proofs and considerations are overly complicated because they assume that the unitary group is not a topological group^{1}. The assertion of proposition 1 has been mentioned in [4] .

The misunderstanding that is not a topological group in the strong topology might come from the fact that the composition map

(7)

is not continuous in the strong topology (where denotes the space of bounded linear operators) and consequently is not a topological group with respect to the strong topology (in the infinite dimensional case). But the restriction of the composition to is continuous since all subsets of are uniformly bounded and equicontinuous.

Another assertion in [3] is that the compact open topology on is strictly stronger than the strong topology^{2} and therefore some efforts are made in [3] to overcome this assumed difficulty. However, again because of the uniform boundedness of the operators in one can show:

Proposition 2: The compact open topology on coincides with the strong topology.

Proof. The compact open topology on and hence on is generated by the seminorms where is compact. Let be a typical neighbourhood of where is compact and. We have to find a strong

neighbourhood of such that. Let. By compactness of

K there is a finite subset such that where is the usual open ball around of radius r. Now, for there exist with and such that. We conclude, for

(8)

As a consequence, the strongly open is contained in. +

Corollary: The group with the strong topology acts continuously by conjugation on the Banach space of compact operators.

This follows from the corresponding result [3] (Appendix 1, A1.1) for the compact open topology or it can be shown as in the proof of Proposition 1 using equicontinuity.

The proof of proposition 2 essentially shows that on an equicontinuous subset W of the strong topology is the same as the compact open topology. Furthermore, both topologies coincide on W with the topology of pointwise convergence on a total subset.

In particular, if is separable with orthonormal basis, the seminorms generate the strong topology. A direct consequence is (in contrast to an assertion in Wikipedia^{3} which explicitly presents with respect to the strong topology as an example of a non-metrizable space):

Proposition 3: The strong topology on is metrizable^{4} if is separable.

The remarkable result of Kuiper [5] that is contractible in the norm topology if is infinite dimensional and separable is true also with respect to the compact open topology (see e.g. [3] ). By proposition 2 we thus have

Corollary: is contractible in the strong topology if is infinite dimensional and separable.

Remark 2. The first three results extend to the projective unitary group: This group is again a topological group in the strong topology, the strong topology coincides with the compact open topology and it is metrizable for separable. Moreover we have the following exact sequence of topological groups

(9)

exhibiting as a central extension of by in the context of topological groups and at the same time as a U(1)-bundle over.

Using the homotopy sequence associated to (9), turns out to be simply connected (with respect to the strong topology). And is an Eilenberg-MacLane space. Hence, is not contractible. (Recall that for natural numbers an Eilenberg-MacLane space is a topological space X whose n^{th} homotopy group is isomorphic to whereas all other homotopy groups are zero.)

The above sequence (9) is not split as an exact sequence of topological groups or as an exact sequence of groups. Moreover, one can show that even a continuous section does not exist [4] : Every section is neither continuous nor a group homomorphism.

3. Search for a Lie Group Structure

In view of the result of proposition 1 it is natural to ask whether has the structure of a Lie group with respect to the strong topology. Let us review what happens in the case of the norm topology:

We know that is a real Banach Lie group in the norm topology: Its local models are open subsets of the space of bounded skew-symmetric operators. L is a real Banach space and a real Lie algebra with respect to the commutator. The exponential map

(10)

is locally invertible and thus provides the manifold structure on the unitary group. In this way, is a Lie group with Lie algebra L.

The same procedure does not work for the strong topology (in the infinite dimensional case). Although it can be shown that the above exponential map is continuous with respect to the strong topologies, it is not a local homeomorphism. A way to see that cannot be a Lie group with local models in L with respect to the strong topology was explained to me by K.-H. Neeb: Choose an orthonormal basis in. The diagonal operators with respect to and contained in form a subgroup which can be identified with the abelian group

(11)

the product of infinitely many circles. The topology on K induced from the strong topology is the product topology. Hence, K is compact. If would be a Lie group in the strong topology then K would be a Lie group as well with models in the space of diagonal operators in L (with the product topology). However, as a compact Lie group K would have to be a finite dimensional manifold.

Note that if exp were locally invertible for the strong topologies then the same would be true for the restriction

(12)

But this restriction is not locally invertible, since for every strong neighbourhood of the inverse image contains all but finitely many straight lines of the form

(13)

where, and exp is not injective on.

According to the importance of in mathematics and physics one might be tempted to use all unitary, strongly continuous one parameter groups

(14)

as the basic geometric and analytic information to find a manifold structure on. Now, Stone’s theorem states that the strongly continuous one parameter groups are exactly the one parameter groups of the following form

(15)

for self adjoint (not necessarily bounded) operators A on. However, the set of all self adjoint operators is not a linear space.

4. Application to Hilbert Bundles

The result of proposition 1 that with the strong topology is a topological group helps to find simpler and more transparent proofs (e.g. than those in [1] and [3] ) and it gives a coherent picture when dealing with fiber bundles or with unitary representations of topological groups. In the following we exemplify the advantage of knowing that is a topological group with respect to the strong topology by applying this result to the study of Hilbert bundles. For a given topological group G the homotopy classification of all equivalence classes of principal fiber bundles over a fixed paracompact space X can be described using the classifying space BG. (Recall that a classifying space BG of a topological group G is the quotient of a weakly contractible space EG (i.e. a topological space for which all its homotopy groups are trivial) by a proper free action of G. It has the property that any G-principal bundle over a paracompact space is isomorphic to a pullback of the principal bundle, and it is unique up to homotopy.) The significance of proposition 1 is that this can be done for or with the strong topology. Let us explain the consequences for the study of Hilbert bundles:

A Hilbert bundle E over a (paracompact) space X is a locally trivial bundle over X with continuous projection π such that the fibers are isomorphic to a separable complex Hilbert space or its projectivation. Here, “isomorphic” means unitarily isomorphic. In particular, this definition requires (in the case of as the typical fiber) that there exists a cover of open subsets with bundle charts (i.e. homeomorphisms)

(16)

such that and

(17)

is unitary for all. Thus, for the bundle E is an ordinary complex vector bundle with typical fiber and structural group.

The transition map for another bundle chart, , is

(18)

completely determined by the projection

(19)

Now, as we have shown above in (3), ψ is continuous, if and only if the induced map

(20)

is strongly continuous. will not be continuous with respect the norm topology, in general. In the case of as the typical fiber of E we have analogous statements.

As a consequence, the natural principal fiber bundle associated to the Hilbert bundle E (the frame bundle with fibers if is the typical fiber) will be a principal fiber bundle whose structural group is with its strong topology and, in general, not with respect to the norm topology. Note that P_{E} will be, in addition, a principal fiber bundle with respect to the norm topology on if and only if there exists an open cover of X with bundle charts such that all the induced transition maps are norm continuous. Let us call such a bundle “norm-defined”.

In the case that is the typical fiber of E (we call such bundles projective Hilbert bundles) we have analogous results for the associated principal bundle P_{E} (with fibers: The structural group is with the strong topology in general. Moreover, whenever E is norm-defined PE can also be viewed as to be a principal fiber bundle with structural group the projective unitary group in its norm topology.

In order to classify the Hilbert bundles over X it is enough to classify the principal fiber bundles with structural groups resp.. Let the set of isomorphism classes of principal fiber bundles with in the norm topology and correspondingly the set of isomorphism classes of principal fiber bundles with in the strong topology. Analogously, we define for.

Unitary group (vector bundles): Since the unitary group is contractible in both topologies every principal bundle P over X is trivial:

(21)

for. (Here, denotes the set of homotopy classes of continuous maps between topological spaces X and Y, and [P] denotes the equivalence class of the principal bundle P over X.) For an arbitrary Hilbert bundle with typical fiber this implies that it is already isomorphic to the trivial bundle For the norm-defined bundles E the associated principal bundle P_{E} is in and an isomorphism can be found which is locally given by transition functions which are induced by norm continuous. Note, that the classifying spaces are weakly contractible for.

Projective unitary group (projective bundles): We know already that is a for both topologies on the projective unitary group which we will indicate by a superscript A. From the homotopy sequence corresponding to the universal bundle

(22)

one concludes that is an Eilenberg-MacLane space. Now, the homotopy classification of principal fiber bundles asserts that there is a bijection between and, the set of homotopy classes of continuous. For a this is cohomology:. We arrive at the following result which is essentially contained in a different form in [3] :

Proposition 4:

・ The isomorphism classes of projective Hilbert bundles over X are in one-to-one correspondence to.

・ The isomorphism classes of norm-defined projective Hilbert bundles over X are also in one-to-one correspondence to where the isomorphisms of the Hilbert bundles are given by norm continuous transition maps.

Note, that the zero element of represents the class of all trivial bundles which also can be described as the class of projective Hilbert bundles E of the form where F is a true vector bundle with fibers.

5. Conclusion

The property of being a topological group serves as a basis for further research in various areas in mathematics and physics where is a symmetry group. Such a research will be supported by that fact, that is, in addition, contractible and metrizable if is infinite dimensional and separable. This has been exemplified in the last part of this paper by deducing the classification of Hilbert bundles from these results concerning the strong topology on.

NOTES

^{1}Explicitly stated in Appendix 1 of [3] .

^{2}In the beginning of Section 2 of [3] and in the Appendix 1.

^{3}See revision (07.11.13):

https://en.wikipedia.org/w/index.php?title=Metrization_theorem&oldid=580602815

^{4}In an earlier version of this note we claimed that is complete. We thank D. Buchholz for pointing out that this is false.

References

[1] Simms, D.J. (1968) Lie Groups and Quantum Mechanics. Lecture Notes in Mathematics 52, Springer, Berlin, 8.

https://doi.org/10.1007/BFb0069914

[2] Bargmann, V. (1954) On Unitary Ray Representations of Continuous Groups. The Annals of Mathematics, 59, 1-46.

https://doi.org/10.2307/1969831

[3] Atiyah, M. and Segal, G. (2004) Twisted K-Theory. Ukr. Mat. Visn., 1, 287-330.

[4] Schottenloher, M. (2008) A Mathematical Introduction to Conformal Field Theory (Lecture Notes in Physics M43). In: Lecture Notes in Physics 759, 2nd Edition. Springer, Berlin.

[5] Kuiper, N. (1964) Contractibility of the Unitary Group in Hilbert Space. Topology, 3, 19-30.

https://doi.org/10.1016/0040-9383(65)90067-4