Strong Limit Theorems in Non-Commutative Probability
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1110 Ryszard Jajte
Strong Limit Theorems in Non-Commutative Probability
Springer-Verlag Berlin Heidelberg New York Tokyo 1985
Author
Ryszard Jajte Institute of Mathematics, University of .r.6dz Banacha 22, 90-238 -t.6dz, Poland
AMS Subject Classification (1980): 46L50, 46L55; 28005, 60F15 ISBN 3-540-13915-X Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13915-X Springer-Verlag New York Heidelberg Berlin Tokyo
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PREFACE Recently many authors have extended a series of fundamental pointwi se convergence theorems in the theory of probabi 1i ty and ergodic theory to the von Neumann algebra context. They have rro v l de d some new tools for mathematical physics and at the same time created interesting techniques in the theory of operator algebras. The main purpose of these notes is to present a selfcontained exposition of some ideas and results from this area. We shall confine ourselves to the case of von Neumann algebras ahd shall not touch on the problems concerning C*algebras. One of the reasons for this is that we are trying to keep the book on a relatively elementary level. The material presented here has been chosen in such a way that only very little knowledge of the theory of operator algebras is needed for reading it. On the other hand, the von Neumann algebras are very natural noncommutative generalizations of Looalgebras, and their rich structure gives the possibilities to obtain the limit theorems in their "almost sure" versions. In a von Neumann algebra one can introduce the "almost uniform" convergence which, in the classical commutative case of the algebra Loo' is equivalent (via Egoroff's theorem) to the almost sure convergence. This type of convergence will be fundamental for the whole book. Recently, C. Lance proved a noncommutative version of the individual ergodic theorem for *automorphisms of a von Neumann algebra. From the point of view of applications in quantum dynamics this result is of great importance. Chapter 2 is devoted to the discussion of some results of this kind and their generalizations. In particular, we prove some "individual" ergodic theorems for normal positive maps of a von Neumann algebra, the noncommutative versions of Kingman's subadditive ergodic theorems for *automorphisms, a random ergodic theorem and a noncommutative local ergodic theorem for quantum dynamical semigroups. Chapter 3 is devoted to the theory of martingales in von Neumann algebras. Conditional expectations in von Neuman
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