On the generators of quantum dynamical semigroups with invariant subalgebras

Abstract

The problem of characterizing GKLS-generators and CP-maps with an invariant von Neumann algebra A appeared in different guises in the literature. We prove two unifying results, which hold even for weakly closed *-algebras: first, we show how to construct a normal form for A-invariant GKLS-generators, if a normal form for A-invariant CP-maps is known — rendering the two problems essentially equivalent. Second, we provide a normal form for A-invariant CP-maps if A is atomic (which includes the finite-dimensional case). As an application we reproduce several results from the literature as direct consequences of our characterizations and thereby point out connections between different fields.