Some considerations on topologies of infinite dimensional unitary coadjoint orbits

The topology of the embedding of the coadjoint orbits of the unitary group U of an infinite dimensional complex Hilbert space H, as canonically determined subsets of the space Ts of symmetric trace-class operators, is investigated. The space Ts is identified with the B-space predual of the Lie-algebra L(H)s of the Lie group U. It is proved, that the orbits consisting of symmetric operators with finite range are (regularly embedded) closed submanifolds of Ts. Such orbits play a role of “generalized phase spaces” of (also nonlinear) quantum mechanics. An alternative method of proving the regularity of the embedding is also given for the “onedimensional” orbit, i.e. for the projective Hilbert space P(H). Closeness of all the orbits lying in Ts is also proved. © 2003 Elsevier B.V. All rights reserved. MSC: 20C99; 57N20; 57N35; 57N40; 81Q70 JGP SC: Lie groups; Quantum mechanics