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INTERACTIONS IN NONCOMMUTATIVE DYNAMICS William Arveson
 

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 well-defined
"state of the past" and a well-defined "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 C-algebra.
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 one-parameter group U = {Ut :
t R} of unitary operators acting on a separable infinite-dimensional Hilbert space

  

Source: Arveson, William - Department of Mathematics, University of California at Berkeley

 

Collections: Mathematics