Orbits of conditional expectations

Abstract

Let N \subseteq M be von Neumann algebras and E:M\to 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 G_M of M has a natural complex analytic structure and the map given by this action: G_M\to 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'\cap M \subseteq 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) < \infty; 2) M is properly infinite and E is any expectation; 3) E is the conditional expectation onto the centralizer of a state. Therefore, in those cases, the fundamental group of U(E) can be characterized as the Weyl group of E.