On the structure of the similarity orbits of Jordan operators as analytic homogeneous manifolds

For Jordan elementsJ in a topological algebraB with unite, an open groupB^–1 of invertible elements and continuous inversion we consider the similarity orbitsS_G(J)={gJg^–1:gG} (G the groupB^–1{e+c:cI},IB a bilateral continuous embedded topological ideal). We construct rational local cross sections to the conjugation mapping and give to the orbitS_G(J) the local structure of a rational manifold. Of particular interest is the caseB=L(H) (bounded linear operators on a separable Hilbert spaceH),I=B, for which we obtain the following:

1.	If for a Hilbert space operator there exist norm continuous local similarity cross sections, then these can be chosen to be rational, especially holomorphic or real analytic.
2.	The similarity orbit of a nice Jordan operator is a rational (especially holomorphic or real analytic) submanifold ofL(H).