Strong Limit Theorems in Noncommutative L2-Spaces
The noncommutative versions of fundamental classical results on the almost sure convergence in L2-spaces are discussed: individual ergodic theorems, strong laws of large numbers, theorems on convergence of orthogonal series, of martingales of powers of co
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen
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Ryszard Jajte
Strong Limit Theorems in Noncommutative L2-Spaces
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Ryszard Jajte Institute of Mathematics L6di University Banacha 22 90-238 Lodz, Poland
Mathematics Subject Classification (1980): 46L50, 46L55, 47A35, 60F15, 81C20
ISBN 3-540-54214-0 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-54214-0 Springer-Verlag New York Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1991 Printed in Germany Printing and binding: Druckhaus Beltz, Hemsbach/Bergstr. 2146/3140-543210 - Printed on acid-free paper
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'PREFACE
This book is a continuation of the volume "Strong limit theorems in non-commutative probability", Lecture Notes in Mathematics 1110 (1985). It is devoted mostly to one subject: the noncommutative versions of pointwise convergence theorems in L2-spaces in the context of von Neumann algebras. In the classical probability and ergodic theory the almost sure convergence theorems for sequences in L2 (over a probability space)
belong to the most important and deep results of these theories. Let us mention here the individual ergodic theorems, the results on the sure convergence of orthogonal series, powers of contractions, martingales and iterates of conditional expectations. The algebraic approach to quantum statistical mechanics suggests the systematic analysis of theorems just mentioned in the context of operator algebras. This is the main goal of this book. We consider a von algebra 1:>1 with a faithful normal state IjJ and take H = L2(M,IjJ) - the completion of
M under the norm
x
x e M.
Then we
introduce a suitable notion of almost sure convergence in H (generalizing the classical one) and prove a series of theorems which can (and should) be treated as the extentions of the well-known classical results (like individual ergodic theorems, Rademacher-Menshov theorem for orthoqonal series or theorem of Burkholder and Chow on the almost sure convergence of the iterates of two conditional expectations etc.). The classical pointwise convergence theorems for sequences in L2 are, as a rule, non-trivial extensions of much easier results concerning the convergence in L2-norm. The same situation is in the noncommutative case. Most of the noncommutative LZ-norm versions of the analogical classical results can be rather ea
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