On the Structure of Covariant Dynamical Semigroups

Dynamical semigroups in B(H) without the norm-continuity assumption are investigated by introducing the notion of form-generator, giving the mathematical expression for the idea of Markovian master equation. It is shown that any form-generator on B(H) admits a standard representation, and the non-uniqueness of this representation is described. The equations for the components of the standard representation of a covariant form-generator are deduced in terms of low-order cohomology of the symmetry group; in particular, noncommutative Levy-Khinchin-type formulae for the shift-covariant form-generators are obtained. Relations with previous works on characterisation of the (unbounded) generator of a dynamical semigroup are discussed.