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Summary: DYNAMICAL INVARIANTS FOR NONCOMMUTATIVE FLOWS
William Arveson
Department of Mathematics
University of California
Berkeley CA 94720, USA
18 November 1995
Abstract. We show that semigroups of endomorphisms of B(H) can often be asso-
ciated 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 mo-
mentum 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.
Contents
Introduction
1. Completely positive semigroups
2. Differential operators on matrix algebras
3. Semielliptic operators
4. Momentum
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