Necessary and sufficient conditions for the conservativity of dynamical semigroups

Dynamical semigroups constitute a quantum-mechanical generalization of Markov semigroups, a concept familiar from the theory of stochastic processes. Let $\mathscr H$ be a Hilbert space and $\mathscr A$ a von Neumann algebra. A dynamical semigroup $P_t$ is a $\sigma$-weakly continuous one-parameter semigroup of completely positive maps of $\mathscr A$ into itself. A semigroup $P_t$ possessing the property of preserving the identity $I\in\mathscr A$ is said to be conservative and its infinitesimal operator $L[\,\cdot\,]$ is said to be regular. The present paper studies necessary and sufficient conditions for strongly continuous dynamical semigroups to be conservative. It is shown that under certain additional assumptions one can formulate necessary and sufficient conditions which are analogous to Feller's condition for regularity of a diffusion process: the equation $P=L[P]$ has no solutions in $\mathscr A_+$. Using a Jensen-type inequality for completely positive maps, constructive sufficient conditions are obtained for conservativeness, in the form of inequalities for commutators. The restriction of a dynamical subgroup to an Abelian subalgebra of $\mathscr L_\infty(\mathbb R^n)$ yields a series of new regularity conditions for both diffusion and jump processes.