 
Summary: INTERACTIONS IN NONCOMMUTATIVE DYNAMICS
William Arveson
Department of Mathematics
University of California
Berkeley CA 94720, USA
4 August, 1999
To the memory of Irving Segal
Abstract. A mathematical notion of interaction is introduced for noncommutative
dynamical systems, i.e., for one parameter groups of automorphisms of B(H) en
dowed with a certain causal structure. With any interaction there is a welldefined
"state of the past" and a welldefined "state of the future". We describe the construc
tion of many interactions involving cocycle perturbations of the CAR/CCR flows and
show that they are nontrivial. The proof of nontriviality is based on a new inequality,
relating the eigenvalue lists of the "past" and "future" states to the norm of a linear
functional on a certain Calgebra.
Introduction, summary of results. In this paper we are concerned with one
parameter groups of automorphisms, of the algebra B(H) of all bounded operators
on a Hilbert space H, which carry a particular kind of causal structure. More
precisely, A history is a pair (U, M) consisting of a oneparameter group U = {Ut :
t R} of unitary operators acting on a separable infinitedimensional Hilbert space
