Detailed balance and quantum dynamical maps

Let T be a stochastic map on a -algebra , and a faithful state. Let be the induced action of T on the GNS Hilbert space , and its adjoint on . We say T obeys detailed balance II if is also induced by a stochastic map. In that case we prove that is a contraction on commuting with the modular operator. The relation of this idea to microscopic reversibility is discussed. An entropy estimate is presented.