Dynamical Invariants for Noncommutative Flows

We show that semigroups of endomorphisms of B(H) can often be associated with a dynamical principle; that is, an infinitesimal structure that completely determines the dynamics. These dynamical invariants are similar to the second order differential equations of classical mechanics, in that one can locate a space of momentum operators, a “Riemannian metric”, and a potential. In the simplest cases these structures occur in n × n matrix algebras, and can be classified in terms of noncommutative geometric invariants. As a consequence, we obtain new information relating to the classification of E0-semigroups acting on type I∞ factors.