Pure E ₀-Semigroups and Absorbing States

Abstract:

An E ₀-semigroup acting on is called pure if its tail von Neumann algebra is trivial in the sense that

We determine all pure E ₀-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 there is convergence to equilibrium in the trace norm

A normal state ω with this property is called an absorbing state for α.

Such E ₀-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.