nLab PU(ℋ)



Group Theory

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



The projective unitary group on an infinite-dimensional separable complex Hilbert space \mathcal{H} is traditionally denoted PU()PU(\mathcal{H}), being the quotient of the unitary group U(ℋ) by its circle subgroup S 1S()S^1 \simeq S(\mathbb{C}) \simeq U(1).




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)

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.


In its operator topology (here), PU()PU(\mathcal{H}) is a well-pointed topological group.

(Hebestreit & Sagave 2020, p. 23, using Dardalat & Pennig 2016, Prop. 2.26)

Homomorphisms into PU()PU(\mathcal{H})

Since Γ\Gamma \,\coloneqq\, PU(ℋ) is not a (finite-dimensional) Lie group, it falls outside the applicability of the general theorem that nearby homomorphisms from compact Lie groups are conjugate. Nevertheless, the conclusion still holds, at least for domain GG a discrete, hence finite group:

The PU(ℋ)-space of homomorphisms Hom(G,PU())Hom\big(G,\, PU(\mathcal{H})\big) is a disjoint union (as here) of orbits of the conjugation action.

This is established in Uribe & Lück 2013, Sec. 15, p. 38.


General discussion:

Concerning well-pointedness:

The /2\mathbb{Z}/2-graded projective unitary group, (reflecting the twists of twisted K-theory not just in degree 3 but also in degree 1):

  • Ellen Maycock Parker, Prop. 2.2 of: The Brauer Group of Graded Continuous Trace C *C^\ast-Algebras, Transactions of the American Mathematical Society 308 1 (1988) (jstor:2000953)

reviewed in

Last revised on September 15, 2022 at 16:26:35. See the history of this page for a list of all contributions to it.