Some considerations on topologies of infinite dimensional unitary coadjoint orbits

TL;DR

This paper studies the topology of coadjoint orbits of the infinite-dimensional unitary group within the space of symmetric trace class operators, establishing conditions for their embedding and manifold structure.

Contribution

It proves that finite-rank symmetric operator orbits form closed submanifolds in the trace class space and offers an alternative proof for the projective Hilbert space case.

Findings

Finite-rank symmetric operator orbits are closed submanifolds.

An alternative proof for the projective Hilbert space orbit.

Technical results on decompositions of projections in generic positions.

Abstract

The topology of the embedding of the coadjoint orbits of the unitary group U(H) of an in-finite dimensional complex Hilbert space H, as canonically determined subsets of the B-space T_s of symmetric trace class operators, is investigated. The space T_s is identified with the B-space predual of the Lie-algebra L(H)_s of the Lie group U(H). It is proved, that orbits con-sisting of symmetric operators with finite rank are (regularly embedded) closed submanifolds of T_s. An alternative method of proving this fact is given for the `one-dimensional' orbit, i.e. for the projective Hilbert space P(H). Also a technical assertion concerning existence of simply related decompositions into one-dimensional projections of two unitary equivalent (orthogonal) projections in their `generic mutual position' is formulated, proved, and illustrated.