Lecture Notes in Mathematics Editors: J.--M. Morel, Cachan F. Takens, Groningen B. Teissier, Paris
1839
3 Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo
Rolf Gohm
Noncommutative Stationary Processes
13
Author Rolf Gohm Ernst-Moritz-Arndt University of Greifswald Department of Mathematics and Computer Science Jahnstr. 15a 17487 Greifswald Germany e-mail:
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Mathematics Subject Classification (2000): 46L53, 46L55, 47B65, 60G10, 60J05 ISSN 0075-8434 ISBN 3-540-20926-3 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag is a part of Springer Science + Business Media http://www.springeronline.com c Springer-Verlag Berlin Heidelberg 2004 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors SPIN: 10983744
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Preface
Research on noncommutative stationary processes leads to an interesting interplay between operator algebraic and probabilistic topics. Thus it is always an invitation to an exchange of ideas between different fields. We explore some new paths into this territory in this book. The presentation proceeds rather systematically and elaborates many connections to already known results as well as some applications. It should be accessible to anyone who has mastered the basics of operator algebras and noncommutative probability but, concentrating on new material, it is no substitute for the study of the older sources (mentioned in the text at appropriate places). For a quick orientation see the Summary on the following page and the Introduction. There are also additional introductions in the beginning of each chapter. The text is a revised version of a manuscript entitled ‘Elements of a spatial theory for noncommutative stationary processes with discrete time index’, which has been written by the author as a habilitation thesis (Greifswald, 2002). It is impossible to give a complete picture of all the mathematical influences on me which shaped this work. I want to thank all who have been engaged in discussions with me. Additionally I want to point out that B. K¨ ummerer and his students C. Hertfelder and T. Lang, sharing some of their conceptions with me in an early stage, influenced the conception of this work. Getting involved with the research of C. K¨ ostler, B.V.R. Bhat, U. Franz and M. Sch¨ urmann broadened my thinking about noncommutative probability. Special thanks to M. Sch¨ urmann for always supporting me in my struggle to find enough time to write. Thanks also to B. K¨ ummerer and to the referees of the original manuscript for many useful remarks and suggestions leading to improvements in the final version. The financial support by the DFG is also gratefully acknowledged.
Greifswald August 2003
Rolf Gohm
Summary
In the first chapter we consider normal unital completely positive maps on von Neumann algebras respecting normal states and study the problem to find normal unital completely positive extensions acting on all bounded operators of the GNS-Hilbert spaces and respecting the corresponding cyclic vectors. We show that there exists a duality relating this problem to a dilation problem on the commutants. Some explicit examples are given. In the second chapter we review different notions of noncommutative Markov processes, emphasizing the structure of a coupling representation. We derive related results on Cuntz algebra representations and on endomorphisms. In particular we prove a conjugacy result which turns out to be closely related to K¨ ummerer-Maassen-scattering theory. The extension theory of the first chapter applied to the transition operators of the Markov processes can be used in a new criterion for asymptotic completeness. We also give an interpretation in terms of entangled states. In the third chapter we give an axiomatic approach to time evolutions of stationary processes which are non-Markovian in general but adapted to a given filtration. We call this an adapted endomorphism. In many cases it can be written as an infinite product of automorphisms which are localized with respect to the filtration. Again considering representations on GNS-Hilbert spaces we define adapted isometries and undertake a detailed study of them in the situation where the filtration can be factorized as a tensor product. Then it turns out that the same ergodic properties which have been used in the second chapter to determine asymptotic completeness now determine the asymptotics of nonlinear prediction errors for the implemented process and solve the problem of unitarity of an adapted isometry. In the fourth chapter we give examples. In particular we show how commutative processes fit into the scheme and that by choosing suitable noncommutative filtrations and adapted endomorphisms our criteria give an answer to a question about subfactors in the theory of von Neumann algebras, namely when the range of the endomorphism is a proper subfactor.
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
Extensions and Dilations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 An Example with 2 × 2 - Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 An Extension Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Weak Tensor Dilations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Equivalence of Weak Tensor Dilations . . . . . . . . . . . . . . . . . . . . . . 1.5 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 The Automorphic Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9 10 13 14 19 21 25 28
2
Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 K¨ ummerer’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Bhat’s Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Coupling Representation on a Hilbert Space . . . . . . . . . . . . . . . . 2.4 Cuntz Algebra Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Cocycles and Coboundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 K¨ ummerer-Maassen-Scattering Theory . . . . . . . . . . . . . . . . . . . . . 2.7 Restrictions and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 An Interpretation Using Entanglement . . . . . . . . . . . . . . . . . . . . .
37 38 42 45 47 52 60 63 68
3
Adaptedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.1 A Motivation: Hessenberg Form of an Isometry . . . . . . . . . . . . . . 74 3.2 Adapted Endomorphisms – An Abstract View . . . . . . . . . . . . . . 79 3.3 Adapted Endomorphisms and Stationary Processes . . . . . . . . . . 86 3.4 Adapted Isometries on Tensor Products of Hilbert Spaces . . . . . 90 3.5 Nonlinear Prediction Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 3.6 The Adjoint of an Adapted Isometry . . . . . . . . . . . . . . . . . . . . . . . 106
VIII
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Contents
Examples and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 4.1 Commutative Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.2 Prediction Errors for Commutative Processes . . . . . . . . . . . . . . . 128 4.3 Low-Dimensional Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 4.4 Clifford Algebras and Generalizations . . . . . . . . . . . . . . . . . . . . . . 136 4.5 Tensor Products of Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 4.6 Noncommutative Extension of Adaptedness . . . . . . . . . . . . . . . . . 144
Appendix A: Some Facts about Unital Completely Positive Maps . . . . . . . . . . . 149 A.1 Stochastic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 A.2 Representation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 A.3 The Isometry v . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 A.4 The Preadjoints C and D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 A.5 Absorbing Vector States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Flow Diagram for the Sections
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