group theory

# Contents

## 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.

## Literature

General discussion:

Concerning well-pointedness:

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