Pure E 0 -semigroups and Absorbing States

An E0-semigroup α = {αt : t ≥ 0} acting on B(H) is called pure if its tail von Neumann algebra is trivial in the sense that ∩tαt(B(H)) = C1. We determine all pure E0-semigroups which have a weakly continuous invariant state ω and which are minimal in an appropriate sense. In such cases the dynamics of the state space must stabilize as follows: for every normal state ρ of B(H) there is convergence to equilibrium in the trace norm lim t→∞ ‖ρ ◦ αt − ω‖ = 0. A normal state ω with this property is called an absorbing state for α. Such E0-semigroups must be cocycle perturbations of CAR/CCR flows, and we develop systematic methods for constructing those perturbations which have absorbing states with prescribed finite eigenvalue lists. 1991 Mathematics Subject Classification. Primary 46L40; Secondary 81E05.