Environment induced decoherence for Markovian evolutions

We study environmental decoherence for a quantum Markov semigroup T acting on an arbitrary von Neumann algebra M. In particular, we analyze the relationships between the decomposition of the algebra induced by decoherence and a sort of "isometric-sweeping decomposition" for the space of states. Moreover, when the semigroup has a faithful normal invariant state, we embed the algebra M in its completion M with respect to the scalar product induced by the faithful state, and we compare the decomposition induced by decoherence with some other kind of isometric-sweeping decompositions of M and M