Some considerations on topologies of infinite dimensional unitary coadjoint orbits

The topology of the embedding of the coadjoint orbits of the unitary group <F>U</F> of an infinite dimensional complex Hilbert space <F>H</F>, as canonically determined subsets of the space <F>Ts</F> of symmetric trace-class operators, is investigated. The space <F>Ts</F> is identified with the <F>B</F>-space predual of the Lie-algebra <F>L(H)s</F> of the Lie group <F>U</F>. It is proved, that the orbits consisting of symmetric operators with finite range are (regularly embedded) closed submanifolds of <F>Ts</F>. 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 <F>P(H)</F>. Closeness of all the orbits lying in <F>Ts</F> is also proved. [Copyright &y& Elsevier]