Abstract
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.
Highlights
The unitary group U ( ) plays an essential role in many areas of mathematics and physics, e.g. in representation theory, number theory, topology and in quantum mechanics
The goal of this paper is to confirm that the unitary group U ( ) 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 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
Summary
The unitary group U ( ) plays an essential role in many areas of mathematics and physics, e.g. in representation theory, number theory, topology and in quantum mechanics. The restriction of the composition to U ( ) × U ( ) is continuous since all subsets of U ( ) are uniformly bounded and equicontinuous Another assertion in [3] is that the compact open topology on U ( ) is strictly stronger than the strong topology and some efforts are made in [3] to overcome this assumed difficulty. Corollary: The group U ( ) 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.
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