Abstract: Minimal E_0 semigroups

\abstract
It is known that every semigroup of normal completely
positive maps of a von Neumann can be ``dilated"
in a particular way to
an E_0-semigroup acting on a larger von Neumann algebra.
The E_0-semigroup is not uniquely determined by the completely
positive semigroup; however, it is unique (up to conjugacy)
provided that certain conditions of {\it minimality}
are met.  Minimality is a subtle property, and it is often
not obvious if it is satisfied
for specific examples even in the simplest case where
the von Neumann algebra is $\Cal B(H)$.

In this paper we clarify these
issues by giving a new characterization of
minimality in terms projective cocycles and their limits.
Our results are valid for semigroups of endomorphisms
acting on arbitrary von Neumann algebras with separable predual.