Spectral properties of Markov semigroups in von Neumann algebras

We investigate spectral properties of Markov semigroups in von Neumann algebras and their dual semigroups in a fairly general setting which assumes only the abelianess of the semigroups and positivity of the maps in question. In particular, we analyse various properties of the spectral subspaces, and relations between the spectra of the Markov semigroup and its dual semigroup. In our analysis, we make extensive use of ergodic and quasi-ergodic projections which seems to be a new but quite fruitful approach.