Detailed balance and quantum dynamical maps

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