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Kuiper's theorem

Contents

Context

Functional analysis

Operator algebra

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

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

Last revised on August 16, 2017 at 03:14:06. See the history of this page for a list of all contributions to it.