nLab Kuiper's theorem



Functional analysis

Operator algebra

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



field theory:

Lagrangian field theory


quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization



States and observables

Operator algebra

Local QFT

Perturbative QFT



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


(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.


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 08:08:33. See the history of this page for a list of all contributions to it.