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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

Throughout, let \mathcal{H} be a separable infinite-dimensional complex Hilbert space.

Theorem

(Kuiper’s theorem)
The topological unitary group U ( ) \mathrm{U}(\mathcal{H}) in either the

or the equivalent

is contractible in that there is a left homotopy between the identity id:U()U()id \;\colon\; U(\mathcal{H}) \to U(\mathcal{H}) and the constant function const e:U()U()const_{\mathrm{e}} \;\colon\; U(\mathcal{H}) \to U(\mathcal{H}).

Similarly, since the general linear group GL()GL(\mathcal{H}) of bounded operators is homotopy-equivalent to U ( ) \mathrm{U}(\mathcal{H}) by the Gram-Schmidt process, it, too, is contractible.

See the commented list of references below.

References

Proof for the norm topology on U(ℋ):

  • Nicolaas Kuiper, Contractibility of the unitary group in Hilbert space, Topology, 3, 19-30 (1964)

  • Luc Illusie, Contractibilité du groupe linéaire des espaces de Hilbert de dimension infinie, Séminaire Bourbaki: années 1964/65 1965/66, exposés 277-312, Séminaire Bourbaki, no. 9 (1966), Exposé no. 284, 9 p. (numdam:SB_1964-1966__9__105_0)

Proof for the strong operator topology on U(ℋ):

Direct proof for the compact-open topology on U(ℋ):

Proof that the compact-open topology agrees with the strong operator topology on U(ℋ), by which contractibility of the former follows from that of the latter:

See also:

Last revised on September 20, 2021 at 04:08:33. See the history of this page for a list of all contributions to it.