Markov shift on non-commutative probability

We consider a class of quantum dissipative semigroup on a von-Neumann algebra which admits a normal invariant state. We investigate asymptotic behavior of the dissipative dynamics and their relation to that of the canonical Markov shift. In case the normal invariant state is also faithful, we also extend the notion of `quantum detailed balance' introduced by Frigerio-Gorini and prove that forward weak Markov process and backward weak Markov process are equivalent by an anti-unitary operator.