Ergodic properties of one-parameter automorphism groups of operator algebras

Abstract

We shall consider the following two types of dynamical systems employed in the foundations of quantum theory, C*-systems and W*-systems. It is widely believed that quantum dynamics is suitably described by one of the two types of systems (C* for quantum spin systems with “short range” interaction, and W* for “long range”) [5, 8, 17, 25, 27, 301. A C*-system is a pair (@, a,), where G! is a C*-algebra (always assumed to contain an identity), and (x, is a strongly continuous one-parameter group of *automorphism. The continuity requirement is 11 a,(a) a II+ 0 for t + 0, a E 0. A W*-system is a pair (M, yI), where M is a W*-algebra, and yI is a point-ultraweakly continuous one-parameter group of *-automorphisms of M. The continuity requirement here is 01, y,(x) x) + 0 for t + 0, x E M and ,u E M, are the a(M, M,)-continuous (=ultraweakly) linear functionals on M. The cone of positive elements in M, is denoted by Mf, and consists of the normal positive functionals on M (states when normalized). Both of the stated continuity requirements are well known to be natural for the respective systems, from the point of view of both mathematics [4, 14, 231, and physics [l, 5, 27, 291. A mathematical scattering theory for the two types of abstract quantum systems does not exist. The closest such seems to be rigorous conditions for return to equilibrium [28]. In this paper we look into some of the foundational problems associated to such a non-commutative scattering theory. Related results have earlier been obtained by Arveson [3,4], Muhly and co-workers [20,22], Kawamura and Tomiyama 1151, and Zsido [37]. Common to these results is the focus on finite systems (i.e., the special case where traces are present). In applications [ 10, 12, 131 it is known, however. that the actual systems are type III, and in this paper we therefore turn to the infinite case. 354 0022-247X/82/060354-19$02.00/0