DECOMPOSITION OF OPERATOR SEMIGROUPS ON W*-ALGEBRAS

We consider semigroups of operators on a W∗-algebra and prove, under appropriate assumptions, the existence of a Jacobs-DeLeeuw-Glicksberg type decomposition. This decomposition splits the algebra into a “stable” and “reversible” part with respect to the semigroup and yields, among others, a structural approach to the Perron-Frobenius spectral theory for completely positive operators on W∗-algebras.