nLab
U(ℋ)

Contents

Context

Group Theory

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

Idea

The unitary group on an infinite-dimensional separable complex Hilbert space \mathcal{H} is traditionally denoted U()U(\mathcal{H}). It does not quite have an established prose name, but is often referred to by some combination of the words “unitary group” and “strong topology”.

Its quotient by the circle subgroup U(1) is the corresponding projective unitary group PU(ℋ).

Definition

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

Definition

The set B()B(\mathcal{H}) of bounded operators carries the following topologies, characterized by the conditions under which a sequence {T kB()} k\{ T_k \,\in\, B(\mathcal{H}) \}_{k \in \mathbb{N}} converges to a fixed operator TB()T \,\in\, B(\mathcal{H}):

The limit of this sequence is limkT k=T\underset{k \to \infty}{lim} T_k \;=\; T

(This list follows Espinoza & Uribe 2014, p. 2.)

Properties

Proposition

The norm topology makes U()U(\mathcal{H}) a topological group, in fact a Banach Lie group.

(e.g. Schottenloher 2013 p. 4/Sec. 3)

Proposition

On U()U(\mathcal{H}) the weak operator topology and strong operator topology agree and make it a topological group.

(Hilgert & Neeb 1993, Cor. 94; Schottenloher 2013, Prop. 1; Espinoza & Uribe 2014, Lem. 1.5)

Proposition

The compact-open topology coincides with the strong operator topology

(Schottenloher 2013 Prop. 2; Espinoza & Uribe 2014, Lem. 1.8)

Proposition

The norm topology on U()U(\mathcal{H}) (from Prop. ) is strictly finer than the (weak or strong) operator topology (from Prop. ).

(e.g. Espinoza & Uribe 2014, p. 5-6, see also Schottenloher 2013, p. 4)

In summary:

(1)U() normU() strongAA=AAU() weakAA=AAU() compop U(\mathcal{H})_{norm} \xrightarrow{ \;\;\; \neq \;\;\; } U(\mathcal{H})_{strong} \overset{ \phantom{AA} = \phantom{AA} }{\leftrightarrow} U(\mathcal{H})_{weak} \overset{ \phantom{AA} = \phantom{AA} }{\leftrightarrow} U(\mathcal{H})_{compop}

(also Espinoza & Uribe 2014, Thm. 1.2)

Proposition

Equipped with the strong topology (Prop. ), U()U(\mathcal{H}) is completely metrizable.

(Neeb 1997, Prop. II.1, Schottenloher 2013 Prop. 3, Espinoza & Uribe 2014, Lem. 1.6)

Proposition

Equipped with the strong topology (Prop. ), U()U(\mathcal{H}) is not locally compact.

(Grigorchuk & de la Harpe 2014, Sec. 5)

Theorem

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

or the

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}).

See the references at Kuiper's theorem.

Proposition

The U(1)-quotient space coprojection of U(ℋ) over PU(ℋ) – both in their strong operator topology – is a circle-principal bundle:

S 1 U() PU()U()/S 1 \array{ S^1 &\hookrightarrow& \mathrm{U}(\mathcal{H}) \\ && \big\downarrow \\ && PU(\mathcal{H}) \mathrlap{ \; \simeq \; \mathrm{U}(\mathcal{H})/S^1 } }

(Simms 1970, Thm. 1)
Remark

Prop. means in particular that U()PU()\mathrm{U}(\mathcal{H}) \xrightarrow{\;} PU(\mathcal{H}) is locally trivial, hence that the coset space coprojection U()U()/S 1\mathrm{U}(\mathcal{H}) \xrightarrow{\;} \mathrm{U}(\mathcal{H})/S^1 admits local sections. See also at coset space coprojection admitting local sections.

References

Early discussion in relation to PU(ℋ):

Proof that the the weak- and strong- operator topologies agree on U()U(\mathcal{H}):

Proof that weak-, strong- as well as the compact-open topology all agree on U()U(\mathcal{H}):

Previous influential but wrong claim that they do not:

Further properties:

Proof of complete metrizability:

Proof of failure of local compactness:

  • Rostislav Grigorchuk, Pierre de la Harpe, Amenability and ergodic properties of topological groups: from Bogolyubov onwards, in: Groups, Graphs and Random Walks, Cambridge University Press 2017 (arXiv:1404.7030, doi:10.1017/9781316576571.011)

Proof of Kazhdan's property (T) for U()U(\mathcal{H}):

Last revised on September 20, 2021 at 04:11:34. See the history of this page for a list of all contributions to it.