Properties of translationally invariant quantum-dynamical semigroups

A large class of translationally invariant quantum-dynamical semigroups on a $C^*$ algebra is considered. Conditions that ensure uniqueness of an invariant state and relaxation to it are found. The properties of the dynamical semigroups which arise in the rigorous derivation of the quantum kinetic equations for a Schrödinger particle in the translationaily invariant case are studied.