 
Summary: Orbits of conditional expectations
M. Argerami and D. Stojanoff
Abstract
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 those
cases, the fundamental group of U(E) can be characterized as the Weyl group
of E.
1. Introduction.
