  In this paper we consider a semigroup of completely positive maps
$\tau=(\tau_t,t \ge 0)$ with a faithful normal invariant state $\phi$ on a
type-$II_1$ factor $\cla_0$ and propose an index theory. We :achieve this via a
more general Kolmogorov's type of construction for stationary Markov processes
which naturally associate a nested isomorphic von-Neumann algebras. In
particular this construction generalizes well known Jones construction
associated with a sub-factor of type-II$_1$ factor.
