Singularization of quantum states in averaging of random unitary channels

We consider the representation of an Euclidean space as an Abelian algebra in the space of quantum states. The Hilbert space of a quantum system is formed as the space of function on the Euclidean space that are square integrable with respect to a shift-invariant measure on the Euclidean space. Let the Euclidean space be equipped with the semigroup (with respect to the convolution) of centered Gaussian measures with trace-class covariation operators. Then the evolution of quantum states can be defined by mean values of random unitary channels of space-shifts on Gaussian random vectors. We prove that either this evolution is quantum dynamical semigroup in the space of quantum state with the invariant subspace of nuclear operators, or images of any quantum state under evolution operators are singular states. We describe values of above singular operators on some abelian subalgebra of multiplication operators.