Some considerations on topologies of infinite dimensional unitary coadjoint orbits

Abstract

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 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 . It is proved, that the orbits consisting of symmetric operators with finite range are (regularly embedded) closed submanifolds of T s . 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 “one-dimensional” orbit, i.e. for the projective Hilbert space P( H) . Closeness of all the orbits lying in T s is also proved.