On dynamical systems preserving weights

The canonical unitary representation of a locally compact separable group arising from an ergodic action of the group on a von Neumann algebra with separable predual preserving a faithful normal semifinite (infinite) weight is weak mixing. On the contrary, there exists a non-ergodic automorphism of a von Neumann algebra preserving a faithful normal semifinite trace such that the spectral measure and the spectral multiplicity of the induced action are respectively the Haar measure (on the unit circle) and. Despite not even being ergodic, this automorphism has the same spectral data as that of a Bernoulli shift.