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 $E$.