Decoherence for positive semigroups on M2(C)

We extend Blanchard and Olkiewicz's definition of decoherence to open quantum systems whose dynamics are described by semigroups of positive (and not necessarily completely positive) operators on B(h). In particular, in the case h = C^2, we completely characterize the decomposition B(h) = M1⊕M2 of B(h) in the sum of a decoherence-free part M1 and of a space M2 on which the semigroup vanishes with time.