Dilations of covariant quantum dynamical semigroups

A theory of stochastic dilation of the Hudson Parthasarathy type of a covariant quantum dynamical semigroup on a C* or von Neumann
algebra is constructed. The additional assumption of symmetry of the semigroup with respect to a semifinte trace allows the use of Hilbert space methods, while covariance with respect to the action of a Lie group on the algebra gives a better control on the domain of the (unbounded) generator.