_{1}

^{*}

The goal of this paper is to confirm that the unitary group U(H) on an infinite dimensional complex Hilbert space is a topological group in its strong topology, and to emphasize the importance of this property for applications in topology. In addition, it is shown that U(H) in its strong topology is metrizable and contractible if H is separable. As an application Hilbert bundles are classified by homotopy.

The unitary group

It is easy to show and well-known that the unitary group

is continuous for the strong topology on

in particular, in case of a left action of a topological group G on

Whenever

with respect to the strong topology on

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

Proposition 1:

Proof. Indeed, the composition

i.e.

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}. The assertion of proposition 1 has been mentioned in [

The misunderstanding that

is not continuous in the strong topology (where

Another assertion in [^{2} and therefore some efforts are made in [

Proposition 2: The compact open topology on

Proof. The compact open topology on

neighbourhood

K there is a finite subset

As a consequence, the strongly open

Corollary: The group

This follows from the corresponding result [

The proof of proposition 2 essentially shows that on an equicontinuous subset W of

In particular, if ^{3} which explicitly presents

Proposition 3: The strong topology on ^{4} if

The remarkable result of Kuiper [

Corollary:

Remark 2. The first three results extend to the projective unitary group

exhibiting

Using the homotopy sequence associated to (9), ^{th} homotopy group

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

In view of the result of proposition 1 it is natural to ask whether

We know that

is locally invertible and thus provides the manifold structure on the unitary group. In this way,

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

the product of infinitely many circles

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

But this restriction is not locally invertible, since for every strong neighbourhood

where

According to the importance of

as the basic geometric and analytic information to find a manifold structure on

for self adjoint (not necessarily bounded) operators A on

The result of proposition 1 that

A Hilbert bundle E over a (paracompact) space X is a locally trivial bundle

such that

is unitary for all

The transition map for another bundle chart

completely determined by the projection

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

is strongly continuous.

As a consequence, the natural principal fiber bundle _{E} will be, in addition, a principal fiber bundle with respect to the norm topology on

In the case that _{E} (with fibers

In order to classify the Hilbert bundles over X it is enough to classify the principal fiber bundles with structural groups

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

for_{E} is in

Projective unitary group (projective bundles): We know already that

one concludes that

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

Note, that the zero element of

The property of

Schottenloher, M. (2018) The Unitary Group in Its Strong Topology. Advances in Pure Mathematics, 8, 508-515. https://doi.org/10.4236/apm.2018.85029