  We investigate some basic questions concerning the relationship between the
restricted Grassmannian and the theory of Banach Lie-Poisson spaces. By using
universal central extensions of Lie algebras, we find that the restricted
Grassmannian is symplectomorphic to symplectic leaves in certain Banach
Lie-Poisson spaces, and the underlying Banach space can be chosen to be even a
Hilbert space. Smoothness of numerous adjoint and coadjoint orbits of the
restricted unitary group is also established. Several pathological properties
of the restricted algebra are pointed out.
