Quantum extensions of dynamical systems and of Markov semigroups

We investigate some particular completely positive maps which admit a stable commutative Von Neumann subalgebra. The restriction of such maps to the stable algebra is then a Markov operator. In the first part of this article, we propose a recipe in order to find a quantum extension of a given Markov operator in the above sense. We show that the existence of such an extension is linked with the existence of a special form of dilation for the Markov operator studied by Attal in \cite{Att1}, reducing the problem to the extension of dynamical system. We then apply our method to the same problem in continuous time, proving the existence of a quantum extension for L\'evy processes. In the second part of this article, we focus on the case where the commutative algebra is isomorphic to $\Acal=l^\infty(1,...,N)$ with $N$ either finite or infinite. We propose a classification of the CP maps leaving $\Acal$ stable, producing physical examples of each classes.