Orbits of conditional expectations

Let N ⊆ M be von Neumann algebras and E: M → N 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 group of GM of M has a natural complex analytic structure and the map given by this action: GM → S(E) is a smooth principal bundle. It is also shown that if N is finite then S(E) admits a Reductive Structure. These results were known previously under the conditions of finite index and N ′∩M ⊆ N, which are removed in this work. Conversely, if the orbit S(E) has an Homogeneous Reductive Structure for every expectation defined on M, then M is finite. For every algebra M and every expectation E, a covering space of the unitary orbit U(E) is constructed in terms of the connected component of 1 in the normalizer of E. Moreover, this covering space is the universal covering in any of the following cases: 1) M is a finite factor and Ind(E) < ∞; 2) M is properly infinite and E is any expectation; 3) E is the conditional expectation onto the centralizer of a state. Therefore, in thos