Orbits of conditional expectations

ARGERAMI, MARTIN; STOJANOFF, DEMETRIO

ILLINOIS JOURNAL OF MATHEMATICS

UNIV ILLINOIS URBANA-CHAMPAIGN

Año: 2001 vol. 45 p. 243 - 263

0019-2082

Let N ⊆ M be von Neumann algebras and let E : M → N be 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 that the map GM → 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 results were previously known under the additional assumptions that the index is finite and N´ ∩ M ⊆ N. Conversely, if the orbit S(E) has a 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 I in the normalizer of E. Moreover, this covering space is the universal covering in each 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 these cases, the fundamental group of U(E) can be characterized as the Weyl group of E.