Orbits of conditional expectations

Abstract

Let $N \subseteq M$ be von Neumann algebras and let $E:M\to N$ be a faithful normal conditional expectation. In this work it is shown that the similarity orbit ${\cal S}(E)$ of $E$ by the natural action of the invertible group of $G_M$ of $M$ has a natural complex analytic structure and that the map $G_M\to {\cal S}(E)$ given by this action is a smooth principal bundle. It is also shown that if $N$ is finite then ${\cal S}(E)$ admits a Reductive Structure. These results were previously known under the additional assumptions that the index is finite and $N'\cap M \subseteq N$. Conversely, if the orbit ${\cal S}(E)$ 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 ${\cal U}(E)$ is constructed in terms of the connected component of $1$ in the normalizer of $E$. Moreover, this covering space is the universal covering in each of the following cases: (1) $\m$ is a finite factor and $\ind < \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 these cases, the fundamental group of ${\cal U}(E)$ can be characterized as the Weyl group of $E$.