nLab PU(ℋ)

Idea

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

Properties

General

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.

Proposition

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

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

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

Since $\Gamma \,\coloneqq\,$ 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 $G$ a discrete, hence finite group:

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

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

Related concepts

• U(ℋ)

• universal equivariant $PU(\mathcal{H})$-bundle

Literature

General discussion:

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

• Bernardo Uribe, Wolfgang Lück, Section 15 of: Equivariant principal bundles and their classifying spaces, Algebr. Geom. Topol. 14 (2014) 1925-1995 (arXiv:1304.4862, doi:10.2140/agt.2014.14.1925)

• Jesus Espinoza, Bernardo Uribe, Topological properties of spaces of projective unitary representations, Rev. Acad. Colombiana Cienc. Exact. Fís. Natur. 40 (2016), no. 155, 337-352 (arXiv:1511.06785, scielo:S0370-39082016000200013, doi:10.18257/raccefyn.317)

Concerning well-pointedness:

• Fabian Hebestreit, Steffen Sagave, p. 23 of: Homotopical and operator algebraic twisted K-theory, Mathematische Annalen 378 (2020) 1021-1059 (arXiv:1904.01872, doi:10.1007/s00208-020-02066-6)

The $\mathbb{Z}/2$-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 $C^\ast$-Algebras, Transactions of the American Mathematical Society 308 1 (1988) (jstor:2000953)

reviewed in

• Alan Carey, Bai-Ling Wang, p. 5 of: Thom isomorphism and Push-forward map in twisted K-theory, Journal of K-Theory 1 2 (2008) 357-393 (arXiv:math/0507414, doi:10.1017/is007011015jkt011)