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Non-commutative symmetric Markov semigroups. (English) Zbl 0761.46051
The quadratic form technique is developed for the study of non- commutative elliptic operators. Symmetric Markov semigroups are constructed on semi-finite $$W^*$$-algebras by means of non-commutative Dirichlet forms. The approach is to exploit the Hilbert algebra and $$L^ p$$-spaces (in the sense of I. E. Segal) associated with a faithful normal semi-finite trace on the algebra. The construction of generators using unbounded derivations has hitherto been closely tied to the problem of establishing that the derivations themselves generate $$C^*$$- automorphism groups. Here a class of derivations called Dirichlet are introduced which yield Markov semigroups without necessarily being automorphism generators. Moreover a powerful armoury of limiting techniques is made available by the use of quadratic forms.
The potential for using Dirichlet forms in the construction of dynamical semigroups was noticed by S. Albeverio and R. Høegh-Krohn in the senventies. However the method was not exploited until recently. In the last few years J.-L. Sauvageot has used Dirichlet forms in constructing the transverse heat semigroups on a Riemannian foliation $$C^*$$-algebra [C. R. Acad. Sci. Paris, Sér. I 310, No. 7, 531-536 (1990; Zbl 0704.57019)], and E. B. Davies and O. S. Rothaus have studied the Bochner Laplacian of a Euclidean connection on a vector bundle by viewing it as the generator of a non-commutative Markov semigroup [J. Funct. Anal. 85, No. 2, 264-286 (1989; Zbl 0694.46049)]. Here the theory is developed from scratch with a $$W^*$$-point-of-view and many examples and applications are treated. The paper includes a summary of the most useful parts of non-commutative $$L^ p$$-theory.

##### MSC:
 46L51 Noncommutative measure and integration 46L53 Noncommutative probability and statistics 46L54 Free probability and free operator algebras 47D07 Markov semigroups and applications to diffusion processes 57R30 Foliations in differential topology; geometric theory
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