Noncommutative Stationary Processes

147 Pages·2004·1.243 MB·English

by  Rolf Gohm (auth.)

1839
Lecture Notes in Mathematics
Editors:
J.--M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
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Berlin
Heidelberg
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Rolf Gohm
Noncommutative
Stationary Processes
1 3
Author
RolfGohm
Ernst-Moritz-ArndtUniversityofGreifswald
DepartmentofMathematicsandComputerScience
Jahnstr.15a
17487Greifswald
Germany
e-mail:[email protected]
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MathematicsSubjectClassification(2000):46L53,46L55,47B65,60G10,60J05
ISSN0075-8434
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Preface
Research on noncommutative stationary processes leads to an interesting in-
terplay between operator algebraicand probabilistic topics. Thus it is always
aninvitationtoanexchangeofideasbetweendifferentfields.Weexploresome
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 mas-
tered the basics of operator algebras and noncommutative probability but,
concentrating on new material, it is no substitute for the study of the older
sources(mentionedinthe textatappropriateplaces).For aquick 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 spa-
tial theory for noncommutative stationary processes with discrete time in-
dex’, which has been written by the author as a habilitation thesis (Greifs-
wald, 2002). It is impossible to give a complete picture of all the mathemat-
ical 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.Ku¨mmerer 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 involvedwith the researchofC.Ko¨stler,B.V.R.Bhat,U.Franz
and M.Schu¨rmann broadened my thinking about noncommutative probabil-
ity. Specialthanks to M.Schu¨rmannfor alwayssupporting me in my struggle
tofindenoughtime towrite.ThanksalsotoB.Ku¨mmererandtothe referees
oftheoriginalmanuscriptformanyusefulremarksandsuggestionsleadingto
improvements in the final version. The financial support by the DFG is also
gratefully acknowledged.
Greifswald Rolf Gohm
August 2003

Summary
Inthefirstchapterweconsidernormalunitalcompletelypositivemapsonvon
Neumann algebras respecting normal states and study the problem to find
normalunital completely positive extensions acting on allbounded operators
oftheGNS-Hilbertspacesandrespectingthecorrespondingcyclicvectors.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 endomor-
phisms.Inparticularweproveaconjugacyresultwhichturnsouttobeclosely
related to Ku¨mmerer-Maassen-scatteringtheory.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 in-
terpretation 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 com-
mutativeprocessesfitintothe scheme andthatby choosingsuitable noncom-
mutative filtrations and adapted endomorphisms our criteria give an answer
toaquestionaboutsubfactorsinthetheoryofvonNeumannalgebras,namely
when the range of the endomorphism is a proper subfactor.

Contents
Introduction................................................. 1
1 Extensions and Dilations .................................. 9
1.1 An Example with 2×2 - Matrices ......................... 10
1.2 An Extension Problem ................................... 13
1.3 Weak Tensor Dilations ................................... 14
1.4 Equivalence of Weak Tensor Dilations...................... 19
1.5 Duality ................................................ 21
1.6 The Automorphic Case................................... 25
1.7 Examples............................................... 28
2 Markov Processes ......................................... 37
2.1 Ku¨mmerer’s Approach ................................... 38
2.2 Bhat’s Approach ........................................ 42
2.3 Coupling Representation on a Hilbert Space ................ 45
2.4 Cuntz Algebra Representations............................ 47
2.5 Cocycles and Coboundaries ............................... 52
2.6 Ku¨mmerer-Maassen-ScatteringTheory ..................... 60
2.7 Restrictions and Extensions............................... 63
2.8 An Interpretation Using Entanglement ..................... 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 Contents
4 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
AppendixA:
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
1.4(cid:2)(cid:4)(cid:4) (cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:3)(cid:3) 1.(cid:1)(cid:1)3(cid:4)(cid:3)(cid:4)(cid:4) (cid:4)(cid:3)(cid:4)(cid:3)(cid:4)(cid:3)(cid:4)(cid:3)(cid:4)(cid:3)1(cid:4)(cid:3).(cid:4)(cid:3)(cid:1)(cid:1)1(cid:4)(cid:3)(cid:4)(cid:1)(cid:3)(cid:4)(cid:3)(cid:1)(cid:4)(cid:3)(cid:1)(cid:4)(cid:3)(cid:1)(cid:4)(cid:3)(cid:1)(cid:4)(cid:5)(cid:5)(cid:1)(cid:4)(cid:1)(cid:4)(cid:1)(cid:4)(cid:1)(cid:4)(cid:1)(cid:4)(cid:1)(cid:4)(cid:1)(cid:4)(cid:1)(cid:4)(cid:1)(cid:2)(cid:2)(cid:6)(cid:6)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:6)(cid:6)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:6)(cid:6) (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:6)(cid:6)(cid:2)(cid:2)
1.(cid:1)(cid:1)5 1.2 2.2 2.(cid:7)(cid:7)3 2.4 2.5(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:5)(cid:8)(cid:8)2.(cid:1)(cid:1)7
(cid:6)(cid:6) (cid:6)(cid:6) (cid:6)(cid:6)
1.(cid:1)(cid:1)6 2.(cid:1)(cid:1)1(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:3)(cid:3) 2.6 2.8
(cid:6)(cid:6) (cid:6)(cid:6) (cid:6)(cid:6)
1.7(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)3(cid:6).(cid:6)1(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)(cid:6)3(cid:6).2(cid:6)(cid:7)(cid:6)(cid:7)(cid:6)(cid:7)(cid:7)(cid:6)(cid:7)(cid:6)(cid:7)(cid:7)(cid:6)(cid:7)(cid:6)(cid:7)(cid:7)(cid:6)34(cid:10)(cid:10)(cid:6)..(cid:1)(cid:1)(cid:6)31(cid:6)(cid:3)(cid:6)(cid:3)(cid:6)(cid:3)(cid:6)(cid:3)(cid:6)(cid:3)(cid:6)(cid:3)(cid:9)(cid:9)(cid:6)(cid:6)433(cid:3)...(cid:3)(cid:1)(cid:1)(cid:1)(cid:1)245(cid:3)(cid:11)(cid:11)(cid:8)(cid:8)(cid:2)(cid:2)(cid:3)(cid:8)(cid:2)(cid:2)(cid:3)(cid:8)(cid:2)(cid:2)(cid:8)(cid:3)(cid:2)(cid:2)(cid:8)(cid:3)(cid:8)(cid:2)(cid:2)(cid:8)(cid:3)(cid:2)(cid:2)(cid:8)(cid:6)(cid:6)(cid:3)(cid:3)(cid:3)(cid:3)(cid:5)(cid:5)(cid:8)43..(cid:1)(cid:1)36(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:2)(cid:3)(cid:3)4.4
4.5 4.6