The role of the atomic decoherence-free subalgebra in the study of quantum Markov semigroups

We show that for a Quantum Markov Semigroup (QMS) with a faithful normal invariant state, atomicity of the decoherence-free subalgebra and environmental decoherence are equivalent. Moreover, we prove that the predual of the decoherence-free subalgebra is isometrically isomorphic to the subspace of reversible states. We also describe, in an explicit and constructive way, the relationship between the decoherence-free subalgebra and the fixed point subalgebra.