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(Kuiper’s theorem)
For $\mathcal{H}$ a separable infinite-dimensional complex Hilbert space, the group of bounded and invertible linear operators $GL(\mathcal{H})$, regarded as a topological group under the norm topology or strong operator topology or weak operator topology, is contractible.
The unitary group $U(\mathcal{H})$, being homotopy equivalent to $GL(\mathcal{H})$ by the Gram-Schmidt process, is also contractible.
The original paper of Kuiper proved this group to be contractible in the norm topology; later Dixmier and Douady proved contractibility for the strong operator topology. Atiyah and Segal note in their paper on twisted K-theory that there is an easy proof of contractibility in the weak operator topology. One major difference in the topologies is that with the operator topology then it is a CW-complex but with the weak topology then it isn’t even an ANR.
Note that on $U(\mathcal{H})$ the strong operator topology coincides with the compact open topology (Schottenloher), and with these topologies $U(\mathcal{H})$ is a topological group (the same is not true for $GL(\mathcal{H})$). In fact more is true: the compact open, strong and weak topologies and their $\ast$-counterparts all agree on $U(\mathcal{H})$, which in this topology is a Polish group (Espinoza-Uribe).
Last revised on August 16, 2017 at 03:14:06. See the history of this page for a list of all contributions to it.