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

Quantum extensions of dynamical systems and of Markov semigroups