Orbits of conditional expectations

Let N subset of or equal to M be von Neumann algebras and let E : M --> N b e a faithful normal conditional expectation. In this work it is shown that the similarity orbit S (E) of E by the natural action of the invertible gro up of G(M) of M has a natural complex analytic structure and that the map G (M) --> S(E) given by this action is a smooth principal bundle. It is also shown that if N is finite then S(E) admits a Reductive Structure. These res ults were previously known under the additional assumptions that the index is finite and N' boolean AND M subset of or equal to N. Conversely, if the orbit S(E) has a Homogeneous Reductive Structure for every expectation defi ned on M, then Al is finite. For every algebra AT and every expectation E, a covering space of the unitary orbit U(E) is constructed in terms of the c onnected component of 1 in the normalizer of E. Moreover, this covering spa ce is the universal covering in each of the following cases: (1) Af is a fi nite factor and Ind(E) < infinity; (2) M is properly infinite and E is any expectation; (3) E is the conditional expectation onto the centralizer of a state. Therefore, in these cases, the fundamental group of U(E) can be cha racterized as the Weyl group of E.