Jones index of a quantum dynamical semigroup

In this paper we consider a semigroup of completely positive maps $\tau =({\tau }_t,t\ge 0)$ with a faithful normal invariant state $\varphi$ on a type-$I{I}_1$ factor ${\text{cla}}_0$ 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.