Table Of ContentGleason's Theorem and Its Applications
Mathematics and Its Applications (East European Series)
Managing Editor:
M. HAZEWINKEL
Centre for Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. BIALYNICKI-BIRULA, Institute of Mathematics, Warsaw University, Poland
H. KURKE, Humboldt University, Berlin, Germany
J. KURZWEIL, Mathematics Institute, Academy of Sciences, Prague, Czechoslovakia
L. LEINDLER, Bolyai Institute, Szeged, Hungary
L. LOV Asz, Bolyai Institute, Szeged, Hungary
D. S. MITRINOVI<J, University of Belgrade, Yugoslavia
S. ROLEWICZ, Polish Academy of Sciences, Warsaw, Poland
BL. H. SENDOV, Bulgarian Academy of Sciences, Sofia, Bulgaria
I. T. TODOROV, Bulgarian Academy of Sciences, Sofia, Bulgaria
H. TRIEBEL, University of lena, Germany
Volume 60
Gleason's Theorem
and Its Applications
Anatolij Dvurecenskij
Mathematical Institute
of the Slovak Academy of Sciences, Bratislava
Czechoslovakia
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Library of Congress Cataloging-in-Publication Data
Dvurechenskij, A. V. (Anatolij Vasifevich), 1949-
Gleason's theorem and its applications/ by A. Dvurecenskij. p. cm. - (Mathematics and
its applications (Kluwer Academic Publishers). East European series; v. 60)
Includes index.
ISBN 978-90-481-4209-5 ISBN 978-94-015-8222-3 (eBook)
DOI 10.1007/978-94-015-8222-3
1. Gleason measures. 2. Spaces. Generalized. 3. Mathematical physics, 1. Title. II. Series.
OCI74;17.G56088 1993
530.1 2-do 20 92-32220
Published in co-edition with
Ister Science Press Ltd., Staromestska. 6, 811 06 Bratislava,
Czechoslovakia
© Anatolij Dvurecenskij, 1993
Originally pub1ished by K1uwer Academic Pub1ishers in 1993
Softcover reprint of the hardcover 1s t edition 1993
All right reserved. No part of the material protected by this copyright notice may be
reproduced or utilized in any form or by any means, electronic or mechanical, including
photocopying, recording or by any information storage and retrieval system without written
permission from the copyright owners.
Contents
Preface Xlii
Introduction 1
CHAPTER 1 Hilbert Space Theory 7
1.1 Inner Product Spaces .................................................... 7
1.1.1 Basic Notions ........................................................ 7
1.1.2 Examples of Inner Product Spaces .................................... 8
1.1.3 Norm of Inner Product Spaces ........................................ 9
1.1.4 Exercises ........................................................... 12
1.2 Maximal Orthonormal Systems and Bases ............................... 12
1.2.1 The Gram-Schmidt Orthogonalization Process ....................... 12
1.2.2 Separable Inner Product Spaces ..................................... 13
1.2.3 Orthonormal Bases ................................................. 15
1.2.4 The Existence of Orthonormal Bases for Incomplete Spaces .......... 17
1.2.5 Isomorphism of Inner Product Spaces ................................ 18
1.2.6 Exercises ........................................................... 19
1.3 Linear Functionals and their Applications ............................... 20
1.3.1 Bilinear Forms ...................................................... 20
1.3.2 Linear Transformations ............................................. 21
1.3.3 Linear Functionals .................................................. 23
1.3.4 Aronszajn Functional Hilbert Spaces ................................ 25
1.3.5 The Tensor Product of Hilbert Spaces ............................... 27
1.3.6 Infinite Tensor Products ............................................. 30
1.3.7 Exercises ........................................................... 32
1.4 Linear Operators ........................................................ 34
1.4.1 The Algebra of Linear Operators .................................... 34
1.4.2 Projectors and Splitting Subspaces .................................. 36
1.4.3 Operations with Orthoprojectors .................................... 37
1.4.4 The Spectrum of an Operator ....................................... 41
VI CONTENTS
1.4.5 Spectral Measures .................................................. .42
1.4.6 Exercises ........................................................... 44
1.5 Special Classes of Operators ............................................ .45
1.5.1 Topologies on Sand B (S) ........................................... 45
1.5.2 Compact Operators ................................................. 47
1.5.3 Hilbert-Schmidt Operators .......................................... 48
1.5.4 The Class of Trace Operators ....................................... 50
1.5.5 Exercises ........................................................... 53
1.6 Unbounded Operators ................................................... 55
1.6.1 Adjoints of Unbounded Operators ................................... 55
1.6.2 Applications of the Spectral Theorem ................................ 57
1.6.3 Closable Bilinear Forms ............................................. 59
1.6.4 Exercises ........................................................... 62
1.7 C* -algebras and W* -algebras ............................................ 63
1.7.1 Introduction to C*-algebras ......................................... 63
1.7.2 von Neumann Algebras ............................................. 66
1. 7.3 Characterization of von Neumann Algebras .......................... 68
1. 7.4 Exercises ........................................................... 69
CHAPTER 2 Theory of Quantum Logics 71
2.1 Elements of Quantum Logics ............................................ 71
2.1.1 Orthomodular Posets ............................................... 77
2.1.2 Observables ......................................................... 77
2.1.3 Compatibility ....................................................... 80
2.1.4 States and Measures ................................................ 81
2.1.5 Examples of Quantum Logics and OMPs ............................ 83
2.1.6 Exercises ........................................................... 87
2.2 Joint Distributions of Observables ....................................... 89
2.2.1 Introduction ........................................................ 89
2.2.2 Type I Joint Distributions for Compatible Observables ............... 90
2.3 Commutator and Partial Compatibility ................................ 97
2.4 Joint Distributions for Non-compatible Observables ..................... 105
2.5 Joint Distributions in L(H) ............................................ 119
2.6 Exercises .............................................................. 127
CHAPTER 3 Gleason's Theorem 129
3.1 Introduction ........................................................... 129
3.2 Proof of Gleason's Theorem ............................................ 131
3.2.1 Frame Functions ................................................... 132
CONTENTS vii
3.2.2 Bounded Frame Functions on H = R3 ............................... 134
3.2.3 Bounded Frame Functions on Hn ................................... 141
3.2.4 Frame Functions on Infinite-Dimensional H ......................... 144
3.2.5 Gleason's Theorem for Infinite-Dimensional H ...................... 148
3.2.6 The Aarnes Decomposition and Charges ............................ 149
3.3 Measures and von Neumann Algebras ................................ . 151
3.3.1 Exercises .......................................................... 152
3.4 Gleason's Theorem for Infinite Measures ................................ 154
3.4.1 Infinite Measures in L(R3) ••••••..•..•..••..•••.••.•.••..••.••..••. 155
3.4.2 Infinite Measures in L(Hn) ......................................... 157
3.4.3 Infinite Measures in L(H) .......................................... 158
3.5 Cardinal Numbers and Gleason's Theorem .............................. 161
3.6 Classification and Examples of Infinite Measures ........................ 164
3.7 Measures Generated by Bilinear Forms ................................. 167
3.7.1 Measures Generating Bilinear Forms ............................... 167
3.7.2 Regular Measures .................................................. 173
3.8 Gleason's Theorem for Signed Measures ................................ 178
3.9 The Hahn and Jordan Decomposition .................................. 182
3.10 Convergences of Gleason Measures .................................... 188
3.10.1 Weak Convergence ......................................... '" .... 188
3.10.2 Uniform Convergence of Charges .................................. 191
3.11 Gleason States on Tensor Products .................................... 192
3.11.1 Product of Gleason States ........................................ 194
3.11.2 General Consistent Systems ....................................... 196
3.11.3 Exercises ......................................................... 200
CHAPTER 4 Gleason's Theorem and Completeness Criteria 203
4.1 Algebraic Completeness Criteria ...................... '" ............... 204
4.1.1 The Amemiya-Araki Theorem ...................................... 204
4.1.2 Splitting Sets Criteria .............................................. 206
4.1.3 Closed Subspaces Criteria .......................................... 209
4.1.4 Further Algebraic Structures and Criteria .......................... 210
4.2 Measure-Theoretic Completeness Criteria ............................... 213
4.2.1 Completely Additive Signed Measure Criteria ..... " ................ 214
4.2.2 Frame Function Criteria .............................................. 219
4.2.3 Signed Measure Completeness Criteria .............................. 221
4.2.4 Gleason's Theorem and /7-Finite Measures .......................... 223
4.2.5 Charge Completeness Criteria ...................................... 226
4.3 Gleason's Theorem and Regular Charges ............................... 228
4.3.1 Regular Jordan Charges on E(S) ................................... 228
viii CONTENTS
4.3.2 Regular Jordan Charges on F(S) and Completeness ................ 231
4.4 Systems of Charges and Completeness .................................. 235
4.4.1 Pure Finitely Additive States ...................................... 235
4.4.2 Finitely Additive States and Completeness ......................... 237
4.4.3 Expectational Functionals and Completeness ....................... 239
4.4.4 Completeness Criteria Survey ...................................... 241
4.4.5 Exercises .......................................................... 243
CHAPTER 5 Applications of Gleason's Theorem 245
5.1 Sum Logics ............................................................ 245
5.1.1 Sum of Observables ................................................ 245
5.1.2 Homomorphisms of Sum Logics .................................... 253
5.2 Orthogonal Vector-Valued Measures .................................... 256
5.2.1 Basic Properties ................................................... 256
5.2.2 Existence of Orthogonal Measures .................................. 257
5.2.3 Morphisms and Orthogonal Measures ............................... 260
5.2.4 Wigner's Theorem ................................................. 261
5.3 Orthogonal Measures and Hilbert Spaces ............................... 266
5.3.1 Orthogonal Measures on L(H) ..................................... 266
5.3.2 Finitely Additive Orthogonal Measures on L(H) .................... 271
5.3.3 Exercises .......................................................... 274
5.4 Gleason's Theorem and Keller Spaces .................................. 276
5.4.1 Introduction ....................................................... 276
5.4.2 Quadratic Spaces .................................................. 278
5.4.3 Keller Spaces ...................................................... 283
5.4.4 Measures and Quadratic Spaces .................................... 284
5.4.5 Gleason Measures and Quadratic Spaces over R(Z) ................. 287
5.4.6 Gleason Measures and Keller Spaces ................................ 290
5.4.7 Exercises .......................................................... 292
Bibliography 293
Index of Symbols 321
Index 325
SERIES EDITOR'S PREFACE IX
'Et moi, ... , si j'avait su comment en One service mathematics has rendered
revenir, je n'y serais point alle'. the human race. It has put common
sense back where it belongs, on the
Jules Verne
topmost shelf next to the dusty canis
ter labelled 'discarded nonsense'.
The series is divergent; therefore we
Eric T. Bell
may be able to do something with it.
O. Heaviside
Mathematics is a tool for thought. A highly necessary tool in a world where both feedback
and nonlinearities abound. Similarly, all kinds of parts of mathematics serve as tools for
other parts and for other sciences.
Applying a simple rewriting rule to the quote on the right above one finds such state
ments as: 'One service topology has rendered mathematical physics ... 'j 'One service logic
has rendered computer science ... 'j 'One service category theory has rendered mathema
tics ... '. All arguably true. And all statements obtainable this way form part of the raison
d'etre of this series.
This series, Mathematics and Its Applications, started in 1977. Now that over one
hundred volumes have appeared it seems opportune to reexamine its scope. At the time I
wrote
"Growing specialization and diversification have brought a host of monographs and text
books on increasingly specialized topics. However, the 'tree' of knowledge of mathe
matics and related fields does not grow only by putting forth new branches. It also
happens, quite often in fact, that branches which were thought to be completely dis
parate are suddenly seen to be related. Further, the kind and level of sophistication of
mathematics applied in various sciences has changed drastically in recent years: measure
theory is used (non-trivially) in regional and theoretical economics; algebraic geometry
interacts with physics; the Minkowsky lemma, coding theory and the structure of water
meet one another in packing and covering theory; quantum fields, crystal defects and
mathematical programming profit from homotopy theory; Lie algebras are relevant to
filtering; and prediction and electrical engineering can use Stein spaces. And in addi
tion to this there are such new emerging subdisciplines as 'experimental mathematics',
'CFD', 'completely integrable systems', 'chaos, synergetics and largescale order', which
are almost impossible to fit into existing classification schemes. They draw upon widely
different sections of mathematics."
By and large, all this still applies today. It is still true that at first sight mathematics seems
rather fragmented and that to find, see, and exploit the deeper underlying interrelations
more effort is needed and so are books that can help mathematicians and scientists do so.
Accordingly MIA will continue to try to make such books available.
If anything, the description I gave in 1977 is now an understatement. To the examples
of interaction areas one should add string theory where Riemann surfaces, algebraic geo
metry, modular functions, knots, quantum field theory, Kac-Moody algebras, monstrous
moonshine (and more) all come together. And to the examples of things which can be
usefully applied let me add the topic 'finite geometry'; a combination of words which
sounds like it might not even exist, let alone be applicable. And yet it is being applied: to
x
statistics via designs, to radar/sonar detection arrays (via finite projective planes), and to
bus connections of VLSI chips (via difference sets). There seems to be no part of (so-called
pure)mathematics that is not in immediate danger of being applied. And, accordingly, the
applied mathematician needs to be aware of much more. Besides analysis and numerics,
the traditional workhorses, he may need all kinds of combinatorics, algebra, probability,
and so on.
In addition, the applied scientist needs to cope increasingly with the nonlinear world
and the extra mathematical sophistication that this requires. For that is where the rewards
are. Linear models are honest and a bit sad and depressing: proportional efforts and results.
It is the nonlinear world that infinitesimal inputs may result in macroscopic outputs (or
vice versa). To appreciate what I am hinting at: if electronics were linear we would have
no fun with transistors and computers; we would have no TV; in fact you would not be
reading these lines.
There is also no safety in ignoring such outlandish things as nonstandard analysis, su
perspace and anticommutingintegration, p-adic and ultrametric space. All three have ap
plications in both electrical engineering and physics. Once, complex numbers were equally
outlandish, but they frequently proved the shortest path between 'real' results. Similarly,
the first two topics named have already provided a number of 'wormhole' paths. There is
no telling where all this is leading - fortunately.
Thus the original scope of the series, which for various (sound) reasons now comprises
five subseries:white (Japan), yellow (China), red (USSR), blue (Eastern Europe) and green
(everything else), still applies. It has been enlarged a bit to include books treating of the
tools from one subdiscipline which are used in others. Thus the series still aims at books
dealing with:
a central concept which plays an important role in several different mathematical and/or
scientific specialization areas;
new applications of the results and ideas from one area of scientific endeavour into
another;
influences which the results, problems and concepts of one field of enquiry have, and
have had, on the development of another.
The shortest path between two truths Never lend books, for no one ever re
in the real domain passes through the turns them; the only books I have in
complex domain. my library are books that other folk
have lent me.
J. Hadamard
Anatole France
La physique ne nous donne pas seule
ment l'occasion de resoudre des prob The function of an expert is not to be
lems ... elle nous fait pressentir la so more right than other people, but to be
lution. wrong for more sophisticated reasons.
H. Poincare David Butler
Bussum, 1992 Michiel Hazewinkel
