  An operator space analysis of quantum stochastic cocycles is undertaken.
These are cocycles with respect to an ampliated CCR flow, adapted to the
associated filtration of subspaces, or subalgebras. They form a noncommutative
analogue of stochastic semigroups in the sense of Skorohod. One-to-one
correspondences are established between classes of cocycle of interest and
corresponding classes of one-parameter semigroups on associated matrix spaces.
Each of these 'global' semigroups may be viewed as the expectation semigroup of
an associated quantum stochastic cocycle on the corresponding matrix space. The
classes of cocycle covered include completely positive contraction cocycles on
an operator system, or C*-algebra; completely contractive cocycles on an
operator space; and contraction operator cocycles on a Hilbert space. As
indicated by Accardi and Kozyrev, the Schur-action matrix semigroup viewpoint
circumvents technical (domain) limitations inherent in the theory of quantum
stochastic differential equations. An infinitesimal analysis of quantum
stochastic cocycles from the present wider perspective is given in a sister
paper.
