nLab
Kuiper's theorem

Context

Functional analysis

Operator algebra

Contents

Statement

Theorem

(Kuiper’s theorem)

For \mathcal{H} a separable infinite-dimensional complex Hilbert space, the group of bounded and invertible linear operators GL()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()U(\mathcal{H}), being homotopy equivalent to GL()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()U(\mathcal{H}) the strong operator topology coincides with the compact open topology (Schottenloher), and with these topologies U()U(\mathcal{H}) is a topological group (the same is not true for GL()GL(\mathcal{H})). In fact more is true: the compact open, strong and weak topologies and their *\ast-counterparts all agree on U()U(\mathcal{H}), which in this topology is a Polish group (Espinoza-Uribe).

References

Revised on August 16, 2017 03:14:06 by David Roberts (129.127.36.227)