An analogue of Hunt's representation theorem in quantum probability

In quantum mechanics certain operator-valued measures are introduced, called instruments, which are an analogue of the probability measures of classical probability theory. As in the classical case, it is interesting to study convolution semigroups of, instruments on groups and the associated semigroups of probability operators, which now are defined on spaces of functions with values in a von Neumann algebra. We consider a semigroup of probability operators with a continuity property weaker than uniform continuity, and we succeed in characterizing its infinitesimal generator under the additional hypothesis that twice differentiable functions belong to the domain of the generator. Such hypothesis can be proved in some particular cases. In this way a partial quantum analogue of Hunt's representation theorem for the generator of convolution semigroups on Lie groups is obtained. Our result provides also a closed characterization of generators of a new class of not norm continuous quantum dynamical semigroups.