nLab U(ℋ)

Idea

The unitary group of unitary operators on an infinite-dimensional separable complex Hilbert space $\mathcal{H}$ is traditionally denoted $U(\mathcal{H})$. 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 $B(\mathcal{H})$ of bounded operators carries the following topologies, characterized by the conditions under which a sequence $\{ T_k \,\in\, B(\mathcal{H}) \}_{k \in \mathbb{N}}$ converges to a fixed operator $T \,\in\, B(\mathcal{H})$:

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

• …in the norm topology or topology of uniform convergence if

  $$\underset{ { x \in \mathcal{H} } \atop { \vert x \vert \leq 1 } }{sup} {\big\vert T_k(x) - T(x) \big\vert} \;\to\; 0 \,;$$

• …in the strong operator topology or topology of pointwise convergence if

  $$\underset{ x \in \mathcal{H} }{\forall} \; T_k(x) \to T(x) \,;$$

• …in the weak operator topology if

  $$\underset{ x, y \in \mathcal{H} }{\forall} \; \big\langle T_k(x), \, y \big\rangle \to \big\langle T_k(x), \, y \big\rangle \,;$$

• …in the compact-open topology if

  uniform convergence above holds for the restrictions $T_k\vert_{C}$ and $T\vert_C$ to any compact subset $C \subset \mathcal{H}$.

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

Properties

Proposition

The norm topology makes $U(\mathcal{H})$ a topological group, in fact a Banach Lie group. Further, it is metrizable, hence paracompact.

(For it being a Banach Lie group cf. Schottenloher 2013 p. 4/Sec. 3. For paracompactness/metrizability cf. Nikolaus, Sachse & Wockel 2013 Sec. 3)

Proposition

On $U(\mathcal{H})$ 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 $U(\mathcal{H})$ (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)$$U(\mathcal{H})_{norm} \xrightarrow{ \;\;\; \neq \;\;\; } U(\mathcal{H})_{strong} \overset{ \phantom{AA} = \phantom{AA} }{\leftrightarrow} U(\mathcal{H})_{weak} \overset{ \phantom{AA} = \phantom{AA} }{\leftrightarrow} U(\mathcal{H})_{comp.open}$$

(also Espinoza & Uribe 2014, Thm. 1.2)

Proposition

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

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

Theorem

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

• norm topology

or the

• strong operator topology

• weak operator topology

• compact-open topology

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

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:

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

(Simms 1970, Thm. 1)

Remark

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

Related concepts

• stable unitary group

• PU(ℋ)

• one-parameter semigroup

• universal equivariant PU(ℋ)-bundle

References

Early discussion of $U(\mathscr{H})$ motivated from quantum mechanics and in relation to PU(ℋ):

• David John Simms, pp. 5 in: Lie Groups and Quantum Mechanics, Lecture Notes in Mathematics 52, Springer (1968) [doi:10.1007/BFb0069914]

• David John Simms: Topological aspects of the projective unitary group, Math. Proc. Camb. Phil. Soc. 68 1 (1970) 57-60 [doi:10.1017/S0305004100001043]

Beware of the false claim in Simms 1968 p. 10 that U(ℋ) is not a topological group in the strong operator topology, corrected by Schottenloher 2013, Espinoza & Uribe 2014, going back to Schottenloher 1995, but repeated by Atiyah & Segal 2004.

The mistake arises because multiplication in $B(\mathscr{H})$ (bounded linear operators) is not generally continuous in the strong operator topology, according to

• Mark Naimark (translated by Leo Boron), §33.2 in: Normed Rings, P. Noordhoff (1959) [ark:/13960/s20rdczxmrt]

whence $GL(\mathscr{H})$ is indeed not a topological group in the strong operator topology, which however does not imply that its subgroup $U(\mathscr{H}) \subset GL(\mathscr{H})$ cannot have that property (cf. Schottenloher 2013 p 2-3).

Proof that $U(\mathcal{H})$ is a topological group in the strong operator topology:

• Martin Schottenloher, Thm. 3.2 in: Geometrie und Symmetrie in der Physik – Leitmotiv der Mathematischen Physik, Springer (1995) [doi:10.1007/978-3-322-89928-6, pdf]

• Martin Schottenloher, Prop. 3.11 in: A Mathematical Introduction to Conformal Field Theory, Lecture Notes in Physics 759, Springer (2008) [doi:10.1007/978-3-540-68628-6, web]

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

• Joachim Hilgert, Karl-Hermann Neeb, Cor. 9.4 in: Lie Semigroups and their Applications. Lecture Notes in Mathematics 1552, Springer (1993) [doi:10.1007/BFb0084640]

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

• Martin Schottenloher: The Unitary Group In Its Strong Topology, Advances in Pure Mathematics 08 05 (2018) [doi:10.4236/apm.2018.85029, arXiv:1309.5891]

• Jesus Espinoza, Bernardo Uribe: Topological properties of the unitary group, JP Journal of Geometry and Topology 16 1 (2014) 45-55 [arXiv:1407.1869, journal]

Previous influential but wrong claim that they do not and that the compact open topology does not even make a topological group (cf. above, concerning Simms 1968 p 10):

• Michael Atiyah, Graeme Segal, p. 39 & appendix of: Twisted K-theory, Ukrainian Math. Bull. 1, 3 (2004) (arXiv:math/0407054, journal page, published pdf)

Further properties:

Proof of complete metrizability for the strong (and hence weak) operator topology:

• Karl-Hermann Neeb, On a Theorem of S. Banach, Journal of Lie Theory 8 (1997) 293–300 (pdf)

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 $U(\mathcal{H})$:

• Bachir Bekka, Kazhdan’s Property (T) for the unitary group of a separable Hilbert space, Geom. funct. anal. 13, 509–520 (2003) [doi:10.1007/s00039-003-0420-0]

• Thomas Nikolaus, Christoph Sachse, Christoph Wockel, A Smooth Model for the String Group, Int. Math. Res. Not. IMRN 16 (2013) 3678-3721 [doi:10.1093/imrn/rns154, arXiv:1104.4288]