Orbits of Conditional Expectations
by
Martin Argerami
Universidad Nacional de La Plata
Coauthors: Demetrio Stojanoff

Let M be a von Neumann algebra, and E a conditional expectation on M. We consider the similarity orbit S(E) of E, that is

S(E)={gE(g-1·g)g-1: g in GL(M)}

In previous works we considered the differentiable structure of this orbit. What we consider in this work is the topological structure of S(E). For every von Neumann algebra M and every expectation E, a covering space of the similarity orbit S(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 E has finite Jones index;

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 S(E) can be characterized as the Weyl group of E. In particular this allows one to produce, for any discrete group G, an algebra M and an expectation E such that S(E) has G as its fundamental group.

Finally, two new characterizations are given for the Weyl group of certain finite index expectations E:M --> N:

1) when the inclusion is irreducible;

2) when N is the fixed algebra of a finite group of outer automorphisms of M.

Date received: March 12, 1999