By Ishiguro M., Sakamoto Y.

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E. of ~d on and the the and A ergodic C*-algebra But [8]. 3 . when of Let the G =ID = X 2 2 is d-dimensional Then is if if A ~ C(rrd), a basis 6 :AF ÷ for A is the any ac- deri- A of is if ~ k ) = in k Now ~x k nl z 1 if easily Ak are an A one that the generators de- faithful ac- for ergodic can a again linear define the combination ~x k A k c center(A). form the z nd d "'" T if C*-algebra if this result action A only of generators following Theorem an if derivations is abelian. g. d {z ~ C I ]z I = i}).

Quant~Gaussian where ~ is abelian. operators on L 2 ( ~ , ~ , ~ ) , and X t, where ~ is is the set of subsets Then we have a classical stochastic process, isomorphisms. Our set-up thus genuinely contains §2. this occurs when ~ is a the elements of ~ and ~ is the measure defined by ~, and ~ of ~ defined by the projections stochastic process, [21]. be a quantum stochastic process, represent them simultaneously In some cases, the classical theory as a special case. Procesges The natural setting for quantum theory is the theory of c*-algebras, various representations the c*-algebra give us algebras of operators.

Phys. 9 G. Lindblad:Commun. Academic Press, London 1976 Inst. H. Poinear~ 32A (1980) 361-375 Inst. H. Poincar~ 35A (1981) 97-103 Inst. H. Poincar~ 38A (1983)187-197 A307 (1982) 279-285 Math. Phys. 48 (1976) 119-130 i0. B. Davies:Rep. Math. Phys. II (1977) 169-188 II. B. Davies:Commun. Math. Phys. 71 (1980) 277-288 12. A. Martin:Nuovo Cimento 30B ~19~5~ 13. B. Phys. 2f7-238 1--7(1980) 249-255 14. T. Kato:Perturbation Theory for Linear Operators, Springer, Berlin 1966 15. H. Haken: Laser theory, in "Handbuch der Physik", vol XXV/2c (S.