A remark on the structure of symmetric quantum dynamical semigroups on von Neumann algebras

We study the structure of the generator of a symmetric, conservative quantum dynamical semigroup, with norm-bounded generator on a von Neumann algebra equipped with a faithful semi-finite trace. For von Neumann algebras with Abelian commutant (i.e. type I von Neumann algebras), we give a necessary and sufficient algebraic condition for the generator of such a semigroup, to be written as a sum of square of self-adjoint derivations of the von Neumann algebra. This generalizes some of the results obtained by Albeverio, Hoegh-Krohn and Olsen(1) for the special case of the finite-dimensional matrix algebras. We also study similar questions for a class of quantum dynamical semigroups with unbounded generators.