Handbook of Quantum Logic and Quantum Structures

HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES QUANTUM STRUCTURES

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HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES QUANTUM STRUCTURES

Edited by

KURT ENGESSER Universitat Konstanz, Konstanz, Germany

DOV M. GABBAY King's College London, Strand, London, UK

DANIEL LEHMANN The Hebrew University of Jerusalem, Jerusalem, Israel

ELSEVIER Amsterdam • Boston • Heidelberg • London • New York • Oxford Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo

Elsevier Radanveg 29, PO Box 2 11, 1000 AE Amsterdam, The Netherlands Linacre House, Jordan Hill, Oxford OX2 8DP, UK First edition 2007 Copyright 02007 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier's Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected] you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions,and selecting Obtainingpermission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury andlor damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made

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FOREWORD

More than a century ago Hilbert posed his unsolved (and now famous), 23 problems of mathematics. When browsing through the Internet recently I found that Hilbert termed his Sixth Problem non-mathematical. How could Hilbert call a problem of mathematics non-mathematical? And what does this problem say? In 1900, Hilbert, inspired by Euclid's axiomatic system of geometry, formulated his Sixth Problem as follows: To find a few physical axioms that, similar to the axioms of geometry, can describe a theory for a class of physical events that i s as large as possible. The twenties and thirties of the last century were truly exciting times. On the one hand there emerged the new physics which we call quantum physics today. On the other hand, in 1933, N. A. Kolmogorov presented a new axiomatic system which provided a solid basis for modern probability theory. These milestones marked the entrance into a new epoch in that quantum mechanics and modern probability theory opened new gates, not just for science, but for human thinking in general. Heisenberg's Uncertainty Principle showed, however, that the micro world is governed by a new kind of probability laws which differ from the Kolmogorovian ones. This was a great challenge to mathematicians as well as to physicists and logicians. One of the responses to this situation was the, now famous, 1936 paper by Garret Birkhoff and John von Neumann entitled "The logic of quantum mechanics", in which they suggested a new logical model which was based on the Hilbert space formalism of quantum mechanics and which we, today, call a quantum logic. G. Mackey asked the question whether every state on the lattice of projections of a Hilbert space could be described by a density operator; and his young student A. Gleason gave a positive answer to this question. Although this was not part of Gleason's special field of interest, his theorem, now known as Gleason's theorem, had a profound impact and is rightfully considered one of the most important results about quantum logics and structures. Gleason's proof was non-trivial. When John Bell became familiar with it, he said he would leave this field of research unless there would be a simpler proof of Gleason's theorem. Fortunately, Bell did find a relatively simple proof of the partial result that there exists no two-valued measure on a three-dimensional Hilbert space. An elementary proof of Gleason's theorem was presented by R. Cooke, M. Keane and W. Moran in 1985. In the eighties and nineties it was the American school that greatly enriched the theory of quantum structures. For me personally Varadarajan's paper and subsequently his book were the primary sources of inspiration for my work together with

vi

Foreword

Gleason's theorem. The theory of quantum logics and quantum structures inspired many mathematicians, physicists, logicians, experts on information theory as well as philosophers of science. I am proud that in my small country, Czechoslovakia and now Slovakia, research on quantum structures is a thriving field of scientific activity. The achievements characteristic of the eighties and nineties are the fuzzy a p proaches which provided a new way of looking at quantum structures. A whole hierarchy of quantum structures emerged, and many surprising connections with other branches of mathematics and other sciences were discovered. Today we can relate phenomena first observed in quantum mechanics to other branches of science such as complex computer systems and investigations on the functioning of the human brain, etc. In the early nineties, a new organisation called International Quantum Structures Association (IQSA) was founded. IQSA gathers experts on quantum logic and quantum structures from all over the world under its umbrella. It organises regular biannual meetings: Castiglioncello 1992, Prague 1994, Berlin 1996, Liptovsky Mikulas 1998, Cesenatico 2001, Vienna 2002, Denver 2004, Malta 2006. In spring 2005, Dov Gabbay, Kurt Engesser, Daniel Lehmann and Jane Spurr had an excellent idea -to ask experts on quantum logic and quantum structures to write long chapters for the Handbook of Quantum Logic and Quantum Structures. It was a gigantic task to collect and coordinate these contributions by leading experts from all over the world. We are grateful to all four for preparing this monumental opus and to Elsevier for publishing it. When browsing through this Handbook, in my mind I am wandering back to Hilbert's Sixth Problem. I am happy that this problem is in fact not a genuinely mathematical one which, once it is solved, brings things t o a close. Rather it has led to a new development of scientific thought which deeply enriched mathematics, the understanding of the foundations of quantum mechanics, logic and the philosophy of science. The present Handbook is a testimony to this fact. Those who bear witness to it are Dov, Kurt, Daniel, Jane and the numerous authors. Thanks to everybody who helped bring it into existence. Anatolij DvureEenskij, President of IQSA July 2006

vii

EDITORIAL PREFACE

There is a wide spread slogan saying that Quantum Mechanics is the most successful physical theory ever. And, in fact, there is hardly a physicist who does not agree with this. However, there is also the reverse. Not only is Quantum Mechanics unprecedently successful but it also raises fundamental problems which are equally unprecedented not only in the history of physics but in the history of science in general. The most profound problems that Quantum Mechanics raises are conceptual in nature. What is the proper interpretation of Quantum Mechanics? This is a question touching on most fundamental issues, and it is, at this stage, safe to say that there is no answer to this question yet on which physicists and philosophers of science could agree. It is, moreover, no exaggeration to say that the problem of the conceptual understanding of Quantum Mechanics constitutes one of the great intellectual puzzles of our time. The topic of the present Handbook is, though related to this gigantic issue, more modest in nature. It can, briefly, be described as follows. Quantum Mechanics owes is tremendous success to a mathematical formalism. It is the mathematical and logical investigation of the various aspects of this formalism that constitutes the topic of the present Handbook. This formalism, the core of which is the mathematical structure of a Hilbert space, received its final elegant shape in John von Neumann's classic 1932 book "Mathematische Grundlagen der Quantenmechanik" . In 1936, John von Neumann published, jointly with the Harvard mathematician Garret Birkhoff, a paper entitled "The logic of quantum mechanics". In the Introduction the authors say: "The object of the present paper is to discover what logical structure we may hope to find in physical theories which, like quantum mechanics, do not conform to classical logic". The idea of the paper, which was as ingenious as it was revolutionary, was that the Hiibert space formalism of Quantum Mechanics displayed a logical structure that could prove useful to the understanding of Quantum Mechanics. Birkhoff and von Neumann were the first to put forward the idea that there is a link between logic and (the formalism of) Quantum Mechanics, and their now famous paper marked the birth of a field of research which has become known as Quantum Logic. The Birkhoff-von Neumann paper triggered, after some time of dormancy, a rapid development of quantum logical research. Various schools of thought emerged. Let us, in this preface, highlight just a few of the milestones in this development.

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Editorial Preface

In hi famous essay "Is logic empirical?" Putnam put forward the view that the role of logic in Quantum Mechanics is similar to that of geometry in the theory of relativity. On this view, logic is as empirical as geometry. Putnam's revolutionary thesis triggered a discussion which was highly fruitful not only for Quantum Logic but for our views on the nature of logic in general. Another school of thought was initiated by Piron's "Axiomatique Quantique". This school, which has become known as the Geneva school, aimed at reconstructing the formalism of Quantum Mechanics from first principles. Piron's student Diederik Aerts continued this in Brussels. The achievements of the GenevaBrussels school are reflected in various chapters written by Aerts and former students of his. In Italy it was Enrico Beltrametti and Maria Luisa Dalla Chiara, just t o mention two names, who founded another highly influential school which is well represented in this Handbook. Highly sophisticated efforts resulted in linking the logic of Quantum Mechanics to mainstream logic. Just to give a flavour of this, let us mention that Nishimura studied Gentzen type systems in the context of Quantum Logic. Abramsky and Coecke in Oxford and Sernadas in Lisbon as well as others established the connection with Categorial Logic and Linear Logic, and the connection with NonMonotonic Logic was made by the editors. Prior to this, another direction of research had focused on the lattice structures relevant to Quantum Logic. Essentially, this field of research was brought to fruition in the USA by the pioneering work of Foulis and Greechie on orthomodular lattices. Moreover, we have to mention the Czech-Slovak school which was highly influential in establishing the vast field of research dealing with the various abstract Quantum Structures, which constitute the topic of a whole volume of this Handbook. Let us in this context just mention the names of Anatolij DvureEenskij and Sylvia Pulmannovi in Bratislava and Pave1 P t i k in Prague. The editors are happy and grateful to have succeeded in bringing together the most eminent scholars in the field of Quantum Logic and Quantum Structures for the sake of the present Handbook. We cordially thank all the authors for their contributions and their cooperation during the preparation of this work. Most of the authors are members of the Internatonal Quantum Structures Association (IQSA). We would like to express our deep gratitude to IQSA and in particular to its President, Professor Anatolij DvureEenskij, for cooperating closely with us during the preparation of this Handbook. The present Handbook is an impressive document of the intellectual achievements which have been reached in the study of the logical and mathematical structures arising from Quantum Mechanics. We hope that it will turn out to be a milestone on the road that will ultimately lead to the solution of one of the great intellectual puzzles of our time, namely the understanding of Quantum Mechanics.

Editorial Preface

ix

Finally, we would like to thank our Handbook and Publications Administrator Mrs Jane Spurr for her excellent and dedicated work on yet another Handbook of Logic. Thanks too, to our publishers Elsevier and especially to Arjen Sevenster and Andy Deelen, without whose efforts the book would not get onto shelves. The Editors: Kurt Engesser, Dov Gabbay and Daniel Lehmann Konstanz, London, and Jerusalem September 2006

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CONTRIBUTORS Enrico Beltrametti Dipartimento di Fisica, Universita' di Genova, via Dodecaneso 33, 16164 Genova, Italy. beltrameQge.infn.it David Buhagiar Department of Mathematics, Faculty of Science,University of Malta, MsidaMSD.06, Malta. david.buhagiarQum.edu.mt Emmanuel Chetcut i Department of Mathematics, Junior College, University of Malta, Msida MSD.05, Malta. emanuel.chetcutiQum.edu.mt Georges Chevalier UniversitC Claude Bernard, bltiment de mathkmatiques, 43 boulevard du 11novembre 1918, 69622 Villeurbanne cedex, France. chevalie(0math.univ-lyonl.fr Ferdinand Chovanec Department of Natural Sciences, Academy of Armed Forces, P.O. Box 45, SK03101 Liptovsky Mikulas, Slovakia chovanecQaoslm.sk Anatolij DvureEenskij Mathematical Institute, Slovak Academy of Sciences, Stefiinikova 49, SK-81473 Bratislava, Slovakia. [email protected] Kurt Engesser University of Konstanz, 78457 Konstanz, Germany. [email protected] David J . Foulis 1 Sutton Court, Amherst, MA 01002, USA. foulis(0math.urnass.edu

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Contributors

Dov M. Gabbay Department of Computer Science, King's College London, Strand, London WC2R 2LS, UK. dov.gabbayQkcl.ac.uk Richard J. Greechie Department of Mathematics and Statistics, Louisiana Tech University, Railroad Avenue, Ruston, LA 71272, USA. greechieQcoes.latech.edu

Stan Gudder Department of Mathematics, University of Denver, 2360 S. Gaylord St, Denver, CO 80208, USA. [email protected] Jan Hamhalter Faculty of Electrical Engineering, Department of Mathematics, Czech Technical University, Technickd 2, 166 27 Prague 6, Czech Republic. [email protected]

John Harding Department of Mathematical Sciences, New Mexico State University, P O Box 30001, Las Cruces, NM 88003-8001, USA. jhardingQnmsu.edu FrantiSek K6pka Department of Engineering Fundamentals, Faculty of Electrical Engineering, University of Zilina, Workplace Liptovsky Mikulas, SK-031101 Liptovsky Mikulas, Slovakia kopkaQlm.utc.sk Daniel Lehmann School of Computer Science and Engineering, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. 1ehmannQcs.huji.ac.il RenC Mayet 13 Rue Saint Antoine, 69003 Lyon, France. rene.mayetQwanadoo.fr Norman D. Megill Boston Information Group, 19 Locke Lane, Lexington, MA 02420, USA. nmQalum.mit.edu

Contributors

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Mirko Navara Center for Machine Perception, Departtment of Cybernetics, Faculty of Electrical Engineering, Czech Technical University, Technicki 2, 166 27 Praha 6, Czech Republic. navara(0math.feld.cvut.cz Mladen PaviEi6 Faculty of Civil Engineering, University of Zagreb, Kariceva 26, POB 217, HR10001 Zagreb, Croatia. mpavicicQgrad.hr Pave1 P t i k Czech Technical University, Faculty of Electrical Engineering, Department of Mathematics, Technicki 2, 16627-Prague 6, Czech Republik. ptakQnath.fe1d.cvut.c~ Sylvia Pulmannovi Mathematical Institute, Slovak Academy of Sciences, Stefiianikova 49, SK-81473 Bratislava, Slovakia. pulmannhat savba.sk

.

Jaroslaw Pykacz Institute of Mathematics, University of Gdarisk, Wita Stwosca 57, 80-952 Gdarisk, Poland pykaczQnath.univ.gda.pl Isar Stubbe Departement Wiskunde en Informatica, University of Antwerp, Middelheimlaan 1, B-2020 Antwerp, Belgium. isar.stubbeQua.ac.be Bart Van Steirteghem Departamento de Matemitica, Instituto Superior TCcnico, 1049-001 Lisboa, Portugal. bvansQnath.ist.ut1.pt Mingsheng Ying State Key Laboratory of Intelligent Technology and Systems, Department of Computer Science and Technology, Tsinghua University, Beijing 100084, P R China. yingmshQnai1.tsinghua.edu.cn

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CONTENTS Foreword Anatolij DvureEenskij

v

Editorial Preface Kurt Engesser, Dov Gabbay and Daniel Lehmann

vii

List of Authors

xi

New Quantum Structures Anatolij DvureEenskij

1

Quantum Structures and Fuzzy Set Theory Jaroslaw Pykacz

55

Algebraic and Measure-theoretic Properties of Classes of Subspaces of an Inner Product Space David Buhagiar, Emmanuel Chetcuti and Anatolij DvureEenskij

75

Quantum Probability Stan Gudder Quantum Logics as Underlying Structures of Generalized Probability Theory Pave1 PtAk and Sylvia Pulmannov6 Quantum Logic and Partially Ordered Abelian Groups David J. Foulis and Richard J. Greechie Quantum Structures and Operator Algebras Jan Hamhalter Constructions of Quantum Structures Mirko Navara D-Posets Ferdinand Chovanec and FrantiSek K6pka

xvi

Contents

Wigner's Theorem and its Generalizations Georges Chevalier Propositional Systems, Hilbert Lattices and Generalized Hilbert 477 Spaces Isar Stubbe and Bart Van Steirteghem Equations and Hilbert Lattices Renk Mayet The Source of the Orthomodular Law John Harding Starting from the Convex Set of States Enrico Beltrametti Quantum Logic and Automata Theory Mingsheng Ying Quantum Logic and Quantum Computation Mladen PaviEi6 and Norman D. Megill Index

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HANDBOOK O F QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM STRUCTURES Edited by K. Engesser, D. M. Gabbay and D. Lehmann O 2007 Elsevier B.V. All rights reserved

1

NEW QUANTUM STRUCTURES Anatolij DvureEenskij

1 INTRODUCTION

About a century ago D. Hilbert presented his famous program for 20th century mathematics at the Second International Mathematical Congress in Paris, 1900. There, he formulated his famous 23 open mathematical problems. This program was to be extremely influential in the development of 20th century mathematics and constituted a source of deep and fundamental ideas. Many of the important problems that occupied mathematicians and specialists in other fields in the 20th century can be traced back to Hilbert's problems, and many of the great and fundamental results we have at our disposal today were in fact stimulated by this program. One of Hilbert's most interesting ideas, which lies between mathematics and physics and is thus in the center of our considerations, is his Sixth Problem: To find a few physical axioms that, similar t o the axzoms of geometry, can describe a theory for a class of physical events that i s as large as possible. The year 2000 was acclaimed by UNESCO as the World Year of Mathematics. By that time almost all of Hilbert's problems had been solved. Therefore, the Clay Institute of Mathematics, Cambridge, Massachusetts, USA, was looking for new open problems in mathematics, and on May 24, 2000, Profs M. Atiyah and J. Tate presented, again in Paris, a new challenge for the 21st century: 7 open problems to be considered the greatest in mathematics. The solution of each of these problems will be awarded 1 000 000 USD. In the new list there are also problems in connection with quantum mechanics, Yang-Mills theory and the hypothesis on mass differences. So, again, we see that physics and in particular quantum mechanics provide a motivation for deep mathematical problems as it was with Hilbert's problems. It is safe to say that Hilbert's Sixth Problem will continue motivating research in quantum structures even at the beginning of the third millennium. We try to present some ideas inspired by Hilbert's Sixth Problem and describe some new directions followed today in research on quantum structures. Classical physics in the nineteenth and twentieth century was, from the logical point of view, governed by the paradigm introduced by G. Boole [Boole, 18541. In fact, the basic structure underlying the logic of classical physics was that of a Boolean algebra.

2

Anatolij Dvureenskij

In the thirties, the situation in both physics and mathematics was very inv, teresting. A.N. Kolmogorov published his fundamental work [ ~ o l m o ~ o r o19331 in which he, for the first time, axiomatized modern probability theory using the framework of Boolean algebras. However, although the Kolmogorovian model turned out to be very important for mathematics, it did not correctly describe the situation that arose in those days in connection with what at that time was called the new physics and what we nowadays call quantum mechanics. This is due to the Heisenberg uncertainty principle [Heisenberg, 19301, which asserts that the position x and the momentum p of an elementary particle cannot be measured simultaneously with arbitrarily prescribed accuracy. If Amp and A,x denote the inaccuracies of the measurement of the momentum p and position x respectively in a state m, then

where h = h/27r and h is Planck's constant. A similar relation holds for the energy-life time

It is interesting to note that we encounter relations similar to the uncertainty relations also in other scientific areas, for example in the social sciences, in psychology, in research on the human brain, in the study of automata. Psychologists know that, when we examine a patient through questions, the patient's psychological stress and the accuracy of his answers are, essentially, connected via (1.1). It was J. von Neumann [von Neumann, 19321 who made the first fundamental step towards a rigorous mathematical framework of the new mechanics as well as an interpretational logic for it. The core of this framework is the concept of a complex separable Hilbert space. In 1936, G. Birkhoff and J. von Neumann published their paper [~irkhoffand von Neumann, 19361 where they showed that the set of assertions about a certain quantum mechanical system has a structure different from that of a Boolean algebra, and they suggested that this structure be a projective geometry. Central to the description a physical system in quantum mechanics is the concept of measurement the description of which in turn relies on the concept of a state. A state is mathemtically represented as a probability measure on the physical system. The system L(H) of all closed subspaces of a real, complex, or quaternionic ~ , in his investigaHilbert space is of particular interest. G. Mackey [ ~ a c k e19631 tion on the mathematical foundations of quantum mechanics posed the following problem: Describe the set of all states on the quantum logic L(H) for a separable real or complex Hilbert space. A.M. Gleason [Gleeason, 19571 published the answer to Mackey7sproblem in 1957, and this result led to decisive progress in research on Hilbert space quantum mechanics. In addition, Gleason's theorem also influenced

New Quantum Structures

3

purely mathematical research and the theorem was extended to many cases and 19931. structures, for an overview on Gleason's theorem see [~vure~enskij, A turbulent development started in the sixties. There appeared many important results which can be found in numerous monographs, e.g. [Beltrametti and Cassinelli, 1981; Birkhoff, 1967, DvureEenskij, 1993, DvureEenskij and Pulmannovb, 2000; Gudder, 1979; Kalmbach, 1983; Maeda, 1980, Piron, 1976; Ptbk and PulmannovL, 1991, Varadarajan, 19681. New structures appeared such as orthomodular posets and orthoalgebras. Today all these structures, which were inspired by attempts at clarifying the mathematical foundations of quantum mechanics, are called quantum structures. In the early nineties an international association called International Quantum Structure Association (IQSA) was founded. It organizes Biennial Congresses where mathematicians, physicists, logicians, experts on quantum information, and philosophers interested in quantum structures can meet. It was in the nineties when the Slovak and Italian schools contributed an important new concept, namely that of a D-poset (difference poset), or equivalently, a weak orthoalgebra. Later the American school came up with the concept of an effect algebra. All these equivalent partial algebraic structures combine algebraic ideas with fuzzy set ideas. It turned out that this framework is suited for treating the space of effect operators of a Hilbert space as well as the important structures called MV-algebras. The concept of an MV-algebra was introduced by C.C. Chang [Chang, 19581 in 1958. These structures generalize two-valued logic, i.e. the logic having the truth values 0 and 1only, to an infinite-valued logic taking values from the whole , showed that there real interval [O,l]. Later in 1986, D. Mundici [ ~ u n d i c i19861 is a natural equivalence between MV-algebras and Abelian unital lattice-ordered groups with a strong unit u in the sense that any MV-algebra is isomorphic to an interval [O,u] of some Abelian unital lattice-ordered group with strong unit u. Today, MV-algebras constitute an important class of substructures of other quantum structures. For example, the compatible part (i.e. "classical one") of a lattice-ordered effect algebra forms an MV-subalgebra. For a nice survey of MV-algebras see [Cignoli et al., 20001. Recently, G. Georgescu and A. Iorgulescu [ ~ e o r ~ e s cand u Iogulescu, 20011 introduced the concept of a pseudo MV-algebra, which is a non-commutative generalization of the concept of an MV-algebra (now also called GMV-algebra). It is interesting that certain psychological processes may be viewed as illustrating such a non-commutative generalization of twevalued reasoning. The following experiment was performed in clinical medicine in connection with the transplantation of human organs. Two groups of people were asked the same two questions: (1) Do you agree to dedicate your organs for medical transplantation after your death? (2) Do you agree to accept organs of a donor, if neccesary? When the order of questions was changed in the second group, the number of positive answers to the first question was higher in the second group than in the first group. Today, there even exists a programming language [Baudot, 20001 based on a non-commutative logic.

Anatolij Dvurdenskij

4

Clearly, quantum mechanics provides an example. Quantum mechanical measurements are in general non-commutative; the result of an experiment may d e pend on the order of the measurements. Consider, for example, a beam of particles which are prepared in a certain state, and which are sent through a sequence of three polarizing filters Fl,Fz,F3. It is well-known that the order of the filters makes in general a difference. For example, let the filter be polarizing in planes perpendicular to the particle beam, such that FI polarizes vertically, Fz horizontally and F3 at a 45' angle. If we place the filters in the order Fl, Fz,F3,then no particles are detected, but in the order Fl,F3,F2,particles are detected; the difference is due to quantum interference. 20021 presented a represenFor pseudo MV-algebras, the author [~vure~enskij, tation via intervals [0,u], where u is a strong unit in an t-group not necessarily Abelian, thus generalizing a famous result of Mundici [Mundici, 19861 for MValgebras. In 2001, A. DvureEenskij and T. Vetterlein introduced the concept of a pseudo effect algebra and proved an analogue of the interval representation for these structures on the assumption of a special version of the Riesz decomposition property. In the present paper we concentrate on these non-commutative quantum struct ures. From the measure-theoretical point of view, the quantum structures mentioned above are important in the sense that on these structures a measure theory as well as a probability theory can be built up. This will be done below. We will see that, in the endeavour of establishing connections between different areas of mathematics we were led to beautiful mathematical constructions serving a useful unifying function. They serve as unifying tools in the study of diverse structures, such as algebraic structures, lattice ordered and partially ordered groups, probability theory, many-valued logic etc. and can be of use in the broader context of quantum mechanics, quantum computation and non-Kolmogorovian probability theory.

2 QUANTUM LOGICS, EFFECT ALGEBRAS AND D-POSETS

2.1

Orthomodular lattices and orthomodular posets

After the publication of the famous paper by Birkhoff and von Neumann [Birkhoff and von Neumann, 19361 there appeared a whole hierarchy of quantum structures. At the beginning there were orthomodular posets. We recall that an orthomodular poset (OMP, for short) is a poset L with two distinguished elements 0 5 1 and a unary operation ', called an orthocomplementation such that, for all a, b E L, we have (i) a" = a; (ii) b' 5 a' whenever a 5 b; (iii) a V a' = 1;

New Quantum Structures

(iv) a V b E L whenever a 5 b'; (v) b = a V (b A a') whenever a 5 b (orthomodular law). We write a I b whenever a 5 b', and we say that a and b are orthogonal or mutually exclusive. If an OMP is a lattice, we call it an orthomodular lattice (OML for short); if it is a a-complete lattice, we called it a quantum logic; if ai exists in L for any sequence of mutually orthogonal elements {G)of L, L is said to be an orthea-complete logic. See [~eltramettiand Cassinelli, 1981; Birkhoff, 1967; DvureEenskij, 1993; DvureEenskij and Pulmannovii, 2000; Gudder, 1979; Kalmbach, 1983; Maeda, 1980; Piron, 1976; Ptiik and Pulmannovii, 1991; Varadarajan, 19681. The most important example of a quantum logic is the quantum logic L ( H ) of a real, complex or quaternionic Hilbert space H , where L(H) consists of all closed subspaces M of H , and M I denotes its orthogonal complement; the partial order is set-theoretical inclusion. This example is of basic importance for so-called Hilbert space quantum mechanics. An equivalent structure is the system P ( H ) of all projectors on H. We say that two elements a and b of an OMP L are compatible if there exist three mutually orthogonal elements a l l bl, c in L such that a = a1Vc and b = bl Vc. We recall that an OMP L is a Boolean algebra iff all elements of L are compatible. This property shows that a system of mutually compatible elements generates a Boolean subalgebra of L. Compatibility is a very important property because the maximal sets of mutually compatible elements, called blocks, form Boolean subalgebras, and every OMP can be covered by blocks. So blocks constitute a locally classical part of an OMP, where the classical Kolmogorovian theory holds. States are, mathematically, represented as probability measures. Recall that a state on an OMP L is a mapping m : L + [0, 11such that m(1) = 1,and m(aVb) = m(a) m(b) whenever a Ib. If m is a-additive on L, i.e., m(Vi aj) = m(ai) for any sequence of mutually orthogonal elements {aj) for which Vi ai is defined in L, we call it a a-additive state. Similarly we define a completely additive state. It is well-known that every Boolean algebra admits a state, in fact plenty of states. For an OML this neesdn't be true. We have examples of stateless OMLs, [~reechie, 19711. An important construction of OMLs is due to Greechie [Greechie, 19681 using the technique pasting blocks. An overview on the latest development , on this topic see [ ~ a v a r a19761. D.J. Foulis and C.H. Randall [Foulii and Randall, 19721 introduced orthoalgebras which are more general than OMPs. We recall that an orthoalgebra is a set L with two distinguished elements O , 1 , and with a partial binary operation + : L x L t Lsuch that f o r a l l a , b , c ~ L w e h a v e

VEl

xi

+

+ + +

+

(OAi) if a b E L, then b + a E L and a b = b + a (commutativity); (OAii) i f b + c E L a n d a + ( b + c ) E L, t h e n a + b ~ L a n d( a + b ) + c ~L, and a (b c) = (a b) + c (associativity); (OAii) for any a E L there is a unique b E L such that a b is defined, and a + b = 1 (orthocomplement ation);

+

+

Anatolij DvureEenskij

6

(OAiv) if a + a is defined, then a = 0 (consistency). If the assumptions of (OAii) are satisfied, we write a + b + c for the element (a b) c = a (b c) in L. We introduce a partial order 5 on L via a 5 b iff a + c = b for some c E L, and we write a Ib 8 a + b exists in L. A state is a mapping m : L + [O,1] such that m(1) = 1 and m(a + b) = m(a) + m(b). For example, if we put a b := a V b whenever a Ib in an OMP, then we see that any OMP is an orthoalgebra.

+ +

+ +

+

2.2 Eflect algebras and D-posets In the early nineties, two former students of mine, F. KGpka and F. Chovanec [KGpka and Chovanec, 19941 introduced a new structure called a diflerence poset, D-poset for short. In these structures difference of comparable elements is a primary notion. A D-poset or a poset with difference is a system (P; I , @ ,0 , l ) consisting of a partially ordered set P with a partially defined binary operation 8 satisfying the following conditions for all a, b, c E P. (Dl) b 8 a is defined if and only if a 5 b; (D2) If a 5 b, then b 8 a 5 band b e ( b 8 a ) = a ; (D3) I f a < b < c , t h e n c 8 b 5 ~ 8 a a n d ( c 8 a ) ~ ( c 8 b ) = b 8 a .

+

+,

If P is a D-poset, then we can define a partial operation, such that a b = c iff c 8 b = a, and (P,+,0 , l ) is an effect algebra. Effect algebras as introduced by D. Foulis and M.K. Bennett [~oulisand Bennett, 19941, see also [Giuntini and Greuling, 19891, with addition as a primary notion, are structures equivalent to D-posets. We recall that an effect algebra is a non-empty set E with two distinguished elements 0, 1, and with a partial binary operation + : E x E + E such that, for all a, b, c E E, we have

+

+

+

+

(EAi) if a b E E, then b a E E and a b = b a (commutativity); (EAii) i f b + c € E and a + ( b + c ) E E , t h e n a + b ~E and ( a + b ) + c ~ E and , a (b c) = (a b) c (associativity); (EAiii) for any a E E there is a unique b E E such that a b is defined, and a b = 1 (orthocomplementation);

+ +

+ +

+

+

+

(EAiv) if 1 a is defined, then a = 0 (zereone law). Let a and b be two elements of an effect algebra E. We say that (i) a is orthogonal to b and write a I b iff a + b is defined in E; (ii) a is less than or equal to b and write a 5 b 8 there exists an element c E E such that a Ic and a c = b (in this case we also write b 2 a and c = b 8 a); (iii) b is the orthocomplement of a iff b is a (unique) element of E such that b I a and a + b = 1 and it is written as a'. It is clear that if E is an effect algebra, then (E, 5,8 , 0 , 1 ) is a D-poset. We recall that every orthoalgebra is an effect algebra, and an effect algebra is an orthoalgebra iff a I a implies a = 0.

+

New Quantum Structures

7

A basic example of effect algebras is the system E(H) of all Hermitian operators on a Hilbert space H between the zero operator and the identity. E(H) contains the system of all projectors, and E(H) is not a lattice. We recall that E(H) is a basic tool in secalled Hilbert space quantum mechanics. All algebraic structures defined above are, nowadays, called quantum structures. Another prototypical example of effect algebras is the following. Let (G, u) be an Abelian unital po-group with a strong unit u, and if

+,

+,

is endowed with the restriction of the group addition then (I'(G, u); 0, u) is an effect algebra. We say that an effect algebra E satisfies (i) the Riesz interpolation property, (RIP) for short, if, for all XI, xz, yl, yz in E, xi 5 yj for all i,j implies there exists an element z E E such that xi 5 z 5 yj for all i, j ; (ii) the Riesz decomposition property, (RDP) for short, if x 5 yl yz implies that there exist two elements X I , xz E E with XI 5 yl and xz 5 yz such that x = XI xz. We recall that (1) if E is a lattice, then E trivially satisfies (RIP); the converse is not true as we see below. (2) E has (RDP) iff, x l +xz = yl y2 implies that there 2 E such that XI = ~ 1 1 ~ 1 25, 2 = C Z ~ CZZ, exist four elements ell, cl2, C Z ~~, 2 E y1 = ell czl, and yz = clz czz, [ ~ v u r e ~ e n s kand i j Pulmannovi, 2000, Lemma 1.7.51. (3) (RDP) implies (RIP), but the converse is not true (e.g. if E = L(H), the system of all closed subspaces of a Hilbert space H , then E is a complete lattice but without (RDP)). On the other hand, every finite poset with (RIP) is a lattice. A partially ordered Abelian group (G; +, 0) is said to satisfy the Riesz decomposition property provided, given x, yl, y2 in GS such that x 5 yl yz, there exist X I , x2 in G+ such that x = x l xz and x j 5 yj for each j = 1,2. This condition is by [Goodearl, 1986, Prop 2.11 equivalent to the following two equivalent conditions:

+

+

+

+ +

+

+

+

+

(a) Given XI, 2 2 , yl, y2 in G such that xi 5 yj for all i,j , there exists z in G such that xi 5 z 5 yj for all i,j . (b) Given xl,x2, yl, y2 in G+such thatx1+x2 = yl+yz, thereexist zll,z12, z21, z22 in G+ such that xi = zil zi2 for each i and yj = z~ z2j for each j .

+

+

According to [Goo], a group G with the %esz decomposition property is said to be an interpo2ation group. It is clear that if (G,u) is a unital interpolation group, then E = I'(G,u) has (RDP) . We recall that by a universal group for an effect algebra E we mean a pair (G, 7) consisting of an additive Abelian group G and a G-valued measure 7 : E -+ G (i.e., ~ ( ab) = ~ ( a ) ~ ( b whenever ) a b is defined in E ) such that the following

+

+

+

lAn element u E G+ is said to be a strong unit for a po-group G , if given an element g E G , there is an integer n 2 1 such that -nu 5 g 5 nu.

8

Anatolij DvureEenskij

conditions hold: (i) 7(E) generates G. (ii) If H is an additive Abelian group and 4 : E -+ H is an H-valued measure, then there is a group homomorphism qS : G + H such that 4 = qS o 7. According to [~oulisand Bennett, 19941, every effect algebra possesses a universal group. Ravindran [Ravindran, 19961 ([Dvure~enskijand PulmannovB, 2000, Theorem 1.17.171) proved the following important result. THEOREM 1. Let E be an effect algebra with the Riesz decomposition property. Then there exists a unital interpolation group (G, u) with a strong unit u such that r ( G , u) i s isomorphic to El and there i s a G-valued injective measure 7 such that (G, 7) i s a universal group for E. One can even prove categorical equivalence. Let REA be the category of effect algebras with the Riesz decomposition property, i.e. whose objects are effect algebras satisfying the %esz decomposition property whose morphisms are the homomorphisms of effect algebras, and let UCGZ be the category of unital Abelian Ggroups with interpolation with the morphisms taken to be the Lhomomorphisms preserving the distinguished strong unit. The mapping r : ULGZ -+ REA defined by (G, u) cr r(G, u), and r(h) = hlr(G, u ) , is a functor. THEOREM 2. The functor r defines a categorical equivalence between the category ULGZ of unital Abelian t-groups with interpolation and the category REA. Moreover, if h i s a morphism of unital l-groups with interpolation, then h is injective if and only if r(h) i s injective, and h i s surjective if and only if r ( h ) is surjective. As a corollary we have that every effect algebra with (RDP) has at least one state (the definition of a state is the same as that for orthoalgebras), moreover, every state on r(G, u) is the restriction of a unique.state on (G, u). We recall that a state on (G, u) is any mapping s : G --+ W such that (i)s(g h) = s(g) s(h), g, h E G, (ii) s(g) 2 0 whenever g 2 0, and (iii) s(u) = 1. We recall that a poset (E; 5 ) is an antilattice if only comparable elements of E have an infimum or a supremum. It is clear that any linearly ordered poset is an antilattice and every finite effect algebra with (RIP) is a lattice. The Ravindran result can also be reformulated in the following form:

+

+

THEOREM 3. Every effect algebra E with ( R D P ) i s a subdirect product of antilattice effect algebras with (RDP), and all existing meets and joins in E are preserved in the subdirect product.

2.3 Perfect efect algebras In this subsection, we define a class of perfect effect algebras and we show that they are categorically equivalent to Abelian po-groups. Let G be a directed Abelian po-group and define the lexicographical product

New Quantum Structures

9

where Z is the group of all integers. Then the element (1,O) is a strong unit in the po-group G(Z) and E(G) := F(G(Z), (1, 011, (2.1) is an effect algebra. Every element a E E(G) is of the form either a = (1, -g) or a = (0, g), where g E G+. In addition, if G is a directed interpolation group, then G(Z) is an interpolation group, [~oodearl,1986, Cor 2.121, and E(G) satisfies the %esz interpolation property. Let a be any element of an effect algebra E and n an integer (n 2 0). We define recursively Oa:=O, l a = a , ( n + l ) a = n a + a , n L 1 ,

+

supposing that n a and n a a are defined in E . An ideal of an effect algebra E is a non-empty subset I of E such that (i) x t: E, y E I, x < y imply x E I, and (ii) ifx,y E I and x + y is defined in E, then x + y E I. An ideal I is said to be a Riesz ideal if, for x E I, a, b E E and x < a b, there exist al, bl E I such that x < a1 bl and a1 5 a and bl 5 b. For example, if E has (RDP), then any ideal of E is Riesz. Let A be a subset of an effect algebra E satisfying (RDP). Then the ideal Io(A) of E generated by A is the set

+

+

We denote by Z(E) the set of all ideals of E; then {0), E E Z(E). We recall that if E is linearly ordered, then Z(E) is linearly ordered with respect to setIzbe two ideals of E and assume a € Il \ I2 theoretical inclusion. Indeed, let 11, and b E I2\ Il. We can assume that, e.g., a b which gives a E 12, a contradiction. An element a is said to be infinitesimal if na is defined in E for any integer n 2 1, and denote by Infinit(E) the set of all infinitesimals of E. Then (i) 0 E Infinit(E), (ii) if b E E, a E Infinit(E) and b 5 a, then b E Infinit(E), and (iii) 1 $ Infinit(E). A proper ideal I of an effect algebra E is said to be mmzmal if is not a proper subset of some proper ideal of E (or equivalently, I is a value of 1). By Zorn's lemma E possesses at least one maximal ideal, and let M ( E ) be the set of all maximal ideals of E . We define the radical of E, Rad(E), by


0 and y > 0. Then u = (1,l) is a strong unit for G. The effect algebra E = r(G, u) is an antilattice having (RIP) and (RDP) but E is not a lattice. THEOREM 16 ([DvureEenskij, 2002al). Every block of an effect algebra E with (RIP) and (DMP) is an effect subalgebra of E which is an MV-algebra. Moreover, any such effect algebra is the set-theoretical union of its blocb. THEOREM 17 ([Dvure~enskij,2002al). Every effect algebra E satisfying (RIP) and (DMP) is a set-theoretical union of MV-algebras. THEOREM 18 ([~vure~enskij, 2002al). Every effect algebra E satisfying (RIP) is the set-theoretical union of blocks which are distributive sublattices of E.

2.5 Visualixation o j eflect algebras with (RDP) In the present subsection, we show that effect algebras with (RDP) and MValgebras can be represented and visualized by a set of automorphisms of an antilattice or a linearly ordered set. Let (R, 5 ) be a non-empty antilattice, and let A(R) be the set of all automorphisms a : R + R preserving the partial order 5. Then A(R) can be converted into a pegroup such that the groupaddition is the composition of automorphisms,

15

New Quantum Structures

the order on A(R) is defined via a 5 P iff (w)a 5 (w)P for all w E R, and the neutral element is the identity function on R. Holland [Holland, 19631 proved the basic result that every &group can be embedded into the &group A(R) for some linearly ordered set 51, and Glass [Glass, 1972, Thm 541 generalized this result to directed po-groups satisfying (RIP) showing that every such a pegroup can be embedded into the po-group A(R) for some antilattice $2. We show that a similar result can also be proved for effect algebras. Namely, we prove that every pseudo effect algebra E satisfying (RDP) can be represented as a set of automorphisms of an antilattice R. Such a representation allows us to present effect algebras in a visualized form which can be useful for a more precise investigation of effect algebras. THEOREM 19 ([Dvure~enskij,2002~1). Every effect algebra E with (RDP) can be represented as a n effect algebra of automorphisms of A(R) for some antilattice set R such that all finite infima and suprema existing in E are preserved. If M is an MV-algebra, then M is an effect algebra with (RDP) and then its visualization has the following form. COROLLARY 20. Every MV-algebra M can be represented as a n MV-algebra of automorphisms of A(R) for some linearly ordered set R. The case of the visualization of pseudo effect algebras will be studied below, Subsection 4.6.

2.6 Loomis-Sikorski theorem for eflect algebras The Loomis-Sikorski theorem plays an important role in the study of Boolean a-algebras, see [Sikorski, 19641. Its generalization to a-complete MV-algebras (i.e. MV-algebras which are a a-complete lattice) was proved independently by 20001 and by Mundici [ ~ u n d i c i19991. , It was shown the author [~vure~enskij, that a-complete MV-algebras are always a-homomorphic images of tribes, i.e., of a-complete MV-algebras of fuzzy sets. In [~uhagiaret aL, 20061, we generalized the Loomis-Sikorski theorem to monotone a-complete effect algebras proving that the latter are a-homomorphic images of effect-tribes which are monotone a-complete effect algebras of [0, 11-valued functions with (RDP). Of crucial help was a delicate result of Choquet [Alf, p. 491 which says that the set of extremal states is always a Baire space (not necessarily compact as in the case of MV-algebras). An effect algebra E is monotone a-complete provided that for every ascending (descending) sequence XI 5 xz 5 . . (xl 2 x2 2 . ) in E which is bounded above (below) in E has a supremum (infimum) in E.

.

.

THEOREM 21. Let (G, u ) be a n interpolation group with strong unit. (1) (G, u) is monotone a-complete if and only if E = r ( G , u) is monotone acomplete.

16

Anatolij DvureEenskij

(2) If E = r(G, u) is monotone a-complete, then G is Archimedean. LEMMA 22. Let E be a monotone a-complete effect algebra with (RDP). Let a, b E E. The following two statements are equivalent. (i) s(a) = 0 for each s E Exts(E) implies a = 0. (ii) s(a) = s(b) for each s E Exts(E) implies a = b. Given a compact convex subset K of a topological vector space, we denote by Aff(K) the collection of all affine continuous functions on K . Of course, Aff(K) is an Archimedean partially ordered group and 1 is its strong unit, which is a subgroup of C(K), the system of all continuous real-valued functions on K . Unlike C(K), the space Aff(K) need not to be a lattice. The following lemma ( [ ~ u n c eet al., 1999, Lem 71, [Wright, 1973, Cor 31) will play a crucial role in our investigation. LEMMA 23. Let {a,} be a monotone descending sequence of nonnegative finctions in Aff(K), and let a(x) = limn a,(x) for any x E K . Then A, a, = 0 in Aff(K) if and only if {x E d K : a(%) > 0) is a meager subset in dK, where d K is the set of extreme points of K. The following representation theorem for monotone a-complete interpolation 16.151. groups with strong unit is from [~oodearl,19861

or

THEOREM 24. Let (G, u) be a nonzero monotone a-complete intelpolation group with strong unit and set A = {f E Aff(S(G, u)) : f (s) E s(G) for all discrete s E Exts(G, u)}. Then the natural mapping 4 : G --+ Aff(S(G,u)) provides an isomorphism of (G, u) onto (A, 1) (as ordered groups with strong unit). REMARK 25. In Theorem 24 we have that A is monotone u-complete and, moreover, if f € Aff(S(G, u)), f 5 4(gi), gj € G, for any integer i 2 l, then there is a g E G such that f 5 4(g) 5 4(gi) for any i, see [Goodearl, 1986, Lem 16.171. Therefore, suprema and infima existing in A are the same as those taken in Aff (S(G, 21)). An effect-tribe on a set a # 0 is any system 7 C [O,l]" such that (i) 1 E 7, (ii) i f f E 7, then 1- f E 7, (iii) if f , g E 7 , f 5 1-g, then f + g € 7, and (iv) for any sequences {f,} of elements of 7 such that f, /' f (pointwisely), then f E 7. It is evident that any effect-tribe is a monotone a-complete effect algebra. We recall that given a family B of [O, 11-valued functions on R there is a minimal effect-tribe, '&(B), on 0 generated by B. The following result is a Loomis-Sikorski theorem for monotone a-complete effect algebras with (RDP). Its proof uses the previous theorems.

17

New Quantum Structures

THEOREM 26 ( [ ~ u h a ~ iet a ral., 20061). Let E be a monotone a-complete effect algebra with (RDP). Then there are a convex space 51, an effect-tribe 7 of [O,l]valued functions on 51 with (RDP) and a a-homomorphism h from 7 onto E. An analogous result for a-complete MV-algebras needs the following notion. We recall that a tribe on 51 # 0 is a collection 7 of fuzzy sets from [O,l]" such that (i) 1 E 7, (ii) i f f E 7,then 1- f E 7,and (iii) if {f,}, is sequence from 7,then min{Cz==lf,, 1) E 7 . The Loomis-Sikorski theorem for a-complete MV-algebras is as follows. 2000; Mundici, 19991). Every a-complete MVTHEOREM 27 ([~vure~enskij, algebra M is a a-homomorphic image of a tribe of fuzzy sets.

2.7 Product egect algebras We now introduce product effect algebras. Using the representation theorem for effect algebras with the Riesz decomposition property we show that they are categorically equivalent to unital perings with interpolation. A product on an effect algebra E = (E; 0,1) is any total binary operation . on E such that, for all a, b, c E E, the following holds: If a b is defined in E, thena.c+b-candc.a+c.bexistin E and

+,

+

and we say that E with a product . is a product eflect algebra, and we write E = (E; +, .,0 , l ) . An element u of a product effect algebra E is said to be a unity, ifa.u=u.a=aforany a E E . A product . on E is (i) associative if (a . b) - c = a . (b . c), a, b, c E E ; (ii) commutative if a . b = b - a, a, b E E . It is worth saying that i f . is a product on E, then (iii) a . 0 = 0 = 0 . a; (iv) i f a s b , thenfor any C E E a, . c s b . c a n d c . a I c . b .

-

+

+

Property (iii) follows easily from the following: a 0 = a . (0 0) = a . 0 a - 0 , and the cancellation property gives a . 0 = 0. Similarly, 0 . a = 0. We recall that every effect algebra E possesses at least one commutative and associative product, namely the zero product, i.e. a . b = 0 for all a, b E E . We recall that a po-ring is a ring (R; .,0) such that (i) (R; 0) is an additive 0 and a . a 0 for any c, a 0. If u is a strong Abelian po-group, and c a unit for R, i.e., for any a E R there is an integer n 1 such that a I nu, and if u - u 5 u, then the effect algebra

>

+,

>

>

>

+,

18

Anatolij DvureEenskij

is a product effect algebra with the product . which is the restriction of the ring product . to E x E; the product . is commutative or associative on r ( R , u) whenever so is . on R. For example, if (W; +, .,0) is the ring of the real numbers, then the standard interval [O,1] := r(W, 1) is a product effect algebra. We now present the two main results of this subsection the proof of which uses the basic representation theorem 2.

+,

THEOREM 28 ([DvureEenskij, 2002dl). Let (E; .,0 , l ) be a product effect algebra with the Riesz decomposition property. T h e n there exists a unique (up t o isomorphism) unital po-ring (R, u) satisfying the Riesz decomposition property with the product - and with u u 5 u such that E r ( R , u) and q5(a . b) = q5(a) . q5(b), where q5 i s a n isomorphism of E onto r ( R , u ) preserving the product .. I f . is commutative or associative, so i s . o n R. Denote by PRUDE^ the category of product effect algebras, i.e. its objects are product effect algebras and its morphisms are homomorphisms of effect algebras also preserving .. We denote by R the category of associative unital interpolation po-rings (R, u) with a distinguished strong unit u such that u . u u, and its morphisms are homomorphisms of po-groups which preserve . and the distindefined guished strong units. We denote by rRthe map from R into by ~ R ( ( R+, ; . , 0 , 5 , u ) ) := (r(R,u); -,O,u), and r ~ ( f := ) f lr(R, u ) . THEOREM 29 ([~vure~enskij, 2002dl). T h e functor defines a categorical equivalence of the category R of unital associative interpolation po-rings with a strong unit u such that u . u 5 u and the category PRODEA of product effect algebras.


n, then a b is not defined in M . (iv) Given a E M I , there is a' E M I such that a' 5 a and nu' is defined in M . ( v ) Mo is a normal and mmimal ideal of M such that Mo + Mo = Mo, Mo = Rad(M) = Infinit(M). (vi) Mo is the unique maximal ideal of M , and M E M . (vii) M admits a unique state, namely s(Mi) = iln for each i = 0,1,. . . ,n. Then Mi = s-'({iln)) for any i = 0,1,. . . ,n. (viii) Let M = (Mh,M i , . . . ,MA) be another representation of M satisfying (a)( d ) , then Mi = Mi for each i = 0,1,. .. ,n.

+

+

+

+

By M we denote the system o f pseudo pseudo MV-algebras M such that every maximal ideal o f M is normal. W e note that the converse statement t o Theorem 55, that is, i f M is a (symmetric) pseudo MV-algebra, M E M , and M admits only one extremal state and this state is (n+ 1)-valued, then M is n-perfect, can be proved. Let us define PBMV,, the system of n-perfect pseudo MV-algebras (PBMV: symmetric n-perfect pseudo MV-algebras), V(PBMV,), the variety generated by all n-perfect pseudo MV-algebras. I t is clear that PBMV = P G M V ~ . Let Vpn and be the variety o f pseudo MV-algebras and the variety o f symmetric pseudo MV-algebras, respectively, satisfying the following two equations

v;~

+

( ( n 1) o x " ) ~= 2 O xn+l,

Anatolij DvureEenskij

30

for every integer p, 1 < p

< n, such that p is not

a divisor of n.

THEOREM 56. V(PQMV,) = Vpn = 7(Vpn), and V(PGMV:)

=

v:~.

An n-perfect pseudo MV-algebra M = (Mo, MI,. . . ,M,) = r(G, u) is said to be strong if there is a E Ml such that (i) a belongs to the commutative center of G, and (ii) nu = 1;such an element a is said to be a strong cyclic element of order n. Moreover, M enjoys unique extraction of roots of 1. Indeed, if there is b E M such that nb = 1, then a b = b a in the corresponding group G, which gives n(a-b)=na-nb=O that i s a = b . For example, En(G) is strong; the element a = (1,O) is that in question. On the other hand, every symmetric 1-perfect pseudo MV-algebra is strong, we take a = 1.

+

+

THEOREM 57. An n-perfect pseudo MV-algebra M is isomorphic to some E,(G) if and only if M is strong. In such a case, G is unique up to isomorphism of l-groups. Let SPQMV, denote the category of strong n-perfect pseudo MV-algebras, where objects are pairs (M, a) with a strong n-perfect pseudo MV-algebra M and a fixed strong cyclic element a E M of order n, and morphisms are pseudo MVhomomorphisms of pseudo MV-algebras preserving fixed strong cyclic elements. SPQMV, is a functor such that (E,(G), (1,O)) is an The mapping En : Q object of the category SPQMP,, and if h is a morphism of l-groups, then En (h) (4 9)

=

(0, h(g)) (i, h(g)) (n, -h(g))

if i = 0, g E G+, if 0 < i < n, g E G, if i = n, g E G+,

is a morphism of strong n-perfect pseudo MV-algebras. THEOREM 58. En is a categorical equivalence of the category Q of l-groups and the category SPQMV, of strong n-perfect pseudo MV-algebras. In addition, suppose that h : (En(G), (1,O)) 4 (E,(H), (1,O)) is a morphism of strong n-perfect pseudo MV-algebras, then there is a unique homomorphism f : G 4 H of l-groups such that h = E,(f), and (i) if h is surjective, so is f ; (ii) if h is an isomorphism, so is f THEOREM 59. Let G be a doubly transitive &-group. Then for the variety V(SPBMVn) generated by SPQMV,, we have V(SPQMV,) = V(En(G)). In particular, an identity holds in every strong n-perfect pseudo MV-algebra if and only if it holds in E,(G).

3.5 Pseudo Lukasiewicx BCK-algebras Lukasiewicz BCK-algebras are studied in [Dvure~enskijand PulmannovL, 20001. They are parts of the positive cones of Ggroups. If they admit a greatest element

New Quantum Structures

31

1, they are equivalent to MV-algebras. In the present subsection, we give a noncommutative generalization of the concept of a Lukasiewicz BCK-algebra called a pseudo Lukasiewicz BCK-algebra. A structure (E; 5, O , O ,O), where (E;

David Buhagiar, Emmanuel Chetcuti and Anatolij DvureEenskij

82

(A @ B ~ = ALs ~ nB )E E(S). ~ Thus ~ B AE ALs exists. Since B AE ALs G ALs , we have A@( B AE ALs) E E(S) and therefore, AVE ( B AE ALs) exists. Si~llilarly one can show that if A c B are in C(S), then AVc ( B Ac A'-") exists and is equal to A @ ( B n ALs). Finally we verify that B = A @ ( B n ALs). Since A B, we have B = B n (A@A1s) G ( ~ A)n@ (B n ALs) = A @ ( ~ rALs). l The converse inclusion is trivial. H Let us now show that (Eq(S),C, {0), S,I)need not be an orthomodular poset, in fact we have the following result. THEOREM 11. Eq(S) is orthomodular iJ and only if, E(S) = Eq(S). Proof. Let S be an inner product space. If E(S) = Eq(S) then Eq(S) is orthomodular since E(S) is always orthomodular by Proposition 10. Now suppose that Eq(S) is orthomodular and let M E Eq(S) \ E(S) for a contradiction. Then there exists a vector x E S which is not in M @ MIS. By Proposition 6, M [XI E Eq(S) and by hypothesis we have

+

+ [XI = M V ((M + [x]) A M I S ) . Since M + [XI # M , we cannot have ((M + [XI)r\ M I S ) = (0) and hence, there exists 0 # y E ( M + [XI)n MLs. Therefore, y = a x + m E MLs for some m E M . M

This implies that x E M @ MIS unless a = 0. The two implications lead to a contradiction and thus E(S) = Eq(S). H

c,

c,

We next show that (F(S), {0), S,V, A) and (W(S), {0), S,V, A) are complete lattices with smallest element (0) and largest element S. F(S) is also an ortholattice with respect to Ibut in general W(S) is not orthocomplemented. We will also see later that in general both families need not be orthomodular. PROPOSITION 12. W(S) and F(S) are complete lattices, where for the meet we = AF and i s given by have A =

Aw

while for the join we have

V M, = span (U M,) iEI

v

iEI

in w(S),

iEI

Mi = (span

(2

Mi))

lsls

in F(s).

Algebraic and Measure-theoretic properties

...

83

Proof. It is not difficult to show that W(S) is a lattice with the meet and join defined above. Now let {Mi I i E I) F(S). First we show that (niEI M i ) l s l s = Mi. For any j E I, (niEI Mi)ls MjLs, and therefore

c

niEI

>

Mj ,

for all j E I

iEI

Consequently

and hence

niEI

niEI

Mi E F(S) and therefore AiEI Mi = Mi. Thus Now (span(UiEI M i ) ) l s l s E F(S) and surely it is an upperbound of the collection {Mi I i E I). Let A E F ( S ) be such that A _> Mi for all i E I . Then

so that A=

2 span (U Mi)

lsls

.

iEI

Hence

Vie, Mi = (span(UiEIMi))lSLS.

rn

Although as already mentioned above, the structure on E(S) and C(S) need not be a (complete) lattice, the following propositions show that the infima and suprema in E(S) (respectively, in C(S)) are the same as in F ( S ) provided that these exist in E(S) (respectively, in C(S)). PROPOSITION 13. Let {Ai I i E I) C E(S). The following statements hold:

AiEIAi If ViEI Ai

(1) If

(2)

AiEI Ai = niEIAi. exists i n E(S), then ViEI Ai = (span(UiEI Ai))lSLS. exists i n E(S), then

Proof.

niEI

(1) Let C = AiEI Ai. Then C C_ Ai. If there is a vector v E S which belongs to (niEI Ai) \ C, then C + [v] E E(S) by Proposition 6. Since C [v] Ai for all i E I , we have that C # AiEI Ai which is a contradiction.

+ c

David Buhagiar, Emmanuel Chetcuti and Anatolij Dvureknskij

84

Then by De Morgan's laws, (2) Suppose that ViEIAi exists in E(S). AiEI A": exists in E(S) and moreover we have

=

niEIA":

and therefore

where for the last equality merely note that

njEIA":

=

By the first part, we have that

AiEIA":

(span(UiEI Ai))ls

PROPOSITION 14. Let {Ai I i E I) 5 C(S). The following statements hold:

AiEI Ai exists i n C(S), then AiEI Ai = niEIAi. ~f Vier Ai exists i n C(S), then Vier Ai = (span(UiErA ~ ) ) ~ s ~ s .

(1) If (2)

rn

Proof. The proof is analogous to the proof of Proposition 13.

How much does the algebraic structure of the above classes tell us about the inner product space S itself? Despite the fact that not every inner product space possesses an orthonormal basis (see [Gudder, 1975]), it is still possible (and useful) to define the orthogonal dimension of an inner product space as the cardinality of any maximal orthonormal system in S. However, in contrast to Hilbert spaces, this orthogonal dimension reveals very little information about the properties of the space itself. We recall the well-known fact that two Hilbert spaces are isomorphic if, and only if, they have the same orthogonal dimension. This is not the case with incomplete inner product spaces. Unless we assume completeness, isomorphism between two inner product spaces can only be proved by exhibiting a bijective unitary transformation between the two spaces. Let us recall that P(S1) is said to be isomorphic to P(S2) when there exists a bijective mapping : P(S1) t P(S2) such that:

+

(11) $(AL%) = (~,b(A))-'-~z, for all A E P(S1); (111) +(A v B) = +(A) v +(B), whenever A, B E P(S1) and A 5 ~ (IV)

+-l

satisfies (I), (11) and (111).

~

~

1

;

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.. .

85

In [ ~ u h a ~ iand a r Chetcuti, 20041 the following question was analyzed: Suppose that we are given two separable inner product spaces S1 and S2over W, such that P(S1) is algebraically equivalent to P(S2) as modular lattices. What can be said about S1 and S2? Using Gleason theorem, it was proved that in this case, S1 and S2are unitarily equivalent. This result was then extended by S. Pulmannovii [~ulmannovii,to appear] to non-separable inner product spaces over the complex field C or the division ring of quaternions W. The proof is based on results in projective geometry. One can look at this as a generalization of the classical Wigner's theorem [Wigner, 19591. THEOREM 15. Let S1 and S2be two inner product spaces over K,where K = W, cC or W. Then the following statements are equivalent: (1) S1 is isomorphic to S2 (as inner product spaces); (2) P(S1) is isomorphic to P(S2) (as modular lattices); (3) C(S1) is isomorphic to C(S2) (as orthomodular posets); (4) E(S1) is isomorphic to E(S2) (as orthomodular posets);

(5) Eq(S1) is isomorphic to Eq(S2) (as orthocomplemented posets); (6) F(Sl) is isomorphic to F(S2) (as complete lattices).

4 ALGEBRAIC COMPLETENESS CRITERIA OF INNER PRODUCT SPACES In this section we characterize complete inner product spaces by the algebraic structures of C(S), E(S), Eq(S), F(S) and W (S). In Subsection 4.1 we develop a completeness criterion which is a modification, given by Ptiik and Weber [Ptiik and Weber, 19771 of the original Amerniya-Araki theorem [Amemiya and Araki, 1966671. In Subsection 4.2, we consider the subsequent and successive strengthening of H. Gross and H. Keller's result that S is complete if, and only if, E(S) is a s Keller, 19771. The lattice properties of E(S) and complete lattice [ ~ r o s and C(S) are also investigated in Subsection 4.2 with respect to the completeness of S [Ptiik and Weber, 20011. In Subsection 4.3 we investigate the lattice properties of Eq(S) and also prove that for an inner product space S with d i m y / ~< CCJ we have Eq(S) = E(S) if, and only if, S is complete [Buhagiar and Chetcuti, to appear]. In Subsection 4.4, we define a modular/dual-modular pair in a lattice L and show that an inner product space S is complete if, and only if, in the lattice W(S) of closed subspaces of S, modularity and dual-modularity are equivalent [Holland, 19691.

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The Amemiya-Aralci Theorem.

Inspired by a paper of Piron iron, 19641, Amemiya and Araki proved that if F(S) is orthomodular, then S is complete [Amemiya and Araki, 1966-671. This result initiated further research on the interplay between the algebraic and metric structure of inner product spaces. By introducing the notion of a weakly atomic orthomodular lattice, Ptkk and Weber arrived to the same conclusion of and Wethe Amemiya-Araki Theorem with a somewhat milder assumption [ ~ t & ber, 19771. DEFINITION 16. Let L be a poset. An element a E L is said to be an atom if the inequality b 5 a for b E L implies either b = 0 or b = a. A lattice L is called atomic if for every c E L, c # 0, there exists an atom a E L such that a 5 c. DEFINITION 17. Let L = (L, V, A, I,0 , l ) be an atomic orthocomplemented lattice. L is said to be weakly atomic orthomodular if the following condition is satisfied: If a , b L~a n d a i s a n atom, t h e n a V b = ( ( a ~~ ) A ~ - ! - Lb. ) v Obviously, any orthomodular lattice is weakly atomic orthomodular. However the notion of being weakly atomic orthomodular is in general strictly weaker than the notion of being orthomodular even for finite atomic orthocomplemented lattices. It is proved in [ ~ t b kand Weber, 19771 that an inner product space is complete if, and only if, F(S) is weakly atomic orthomodular. Let us prove this result. PROPOSITION 18. A n inner product space S i s complete if, and only iJ for every x, y E 3 satisfying s l y there exists A E E(S) such that x E 2 and y E -. Proof. We show that the condition is sufficient for the completeness of S. Let x E S\ S be a unit vector. There exists h E S such that (h, a) = 1. Put y = x - h. Then y l x and therefore by hypothesis, there exists a subspace A E E(S) such that x E 2 and y E G.Thus

h = hl + h 2 , where hl E A and h2 E ALs; and

which implies that hl completes the proof.

=x

and h2 = -y, and therefore x E S. The contradiction

PROPOSITION 19. Let M be a complete subspace o f 3 such that dim M T h e n M n S i s dense in M .


112. Let x be any element of M . For any E > 0 there exists y E S such that ))x- yll < €13. Since x E M, we have I(Y,~)I = I(Y-x,e)l < €13.

Algebraic and Measure-theoretic properties

...

Let u = y - u h . Since (u, e) = 0, we have u E S n M. Now (be)

To prove the result for the general finite dimensional case, we apply the process . that M n S is dense in M when of induction on the dimension of M ~ YSuppose dim MLy 5 n - 1. Consider the case when M'-T = span{el, e2, . . .,en). We have

Let N = M @ [el]. Then 3 = N @ span{e2, e3,. . . ,en) and by the induction hypothesis, N r l S = N. Since M is a complete subspace of N and codimension of M in N is one, we have (NnS)nM=M. Consequently, S n M

= M,

that is S n M is dense in M .

rn

20. Let {vl, v2, . . . ,v,) be a finite set of orthonormal vectors in S. Then there exist sequences {upi I i E N), p = l , 2 , . . . ,n, in S such that

PROPOSITION -

lim upi = up ; 2

(upi, vq) = 0 , for each i E N when p # q ; (upi, uqj) = 0 , for each i,j E N when p # q

Proof. Let Hll = span{v2, 213,. .. ,vn)lg. By Proposition 19, there exists a unit vector ull E HllnSsuch that IIv1-u1111 < 112. Set Hzl = span{u11, VI, v3,. . . , v n ) l r and hence, again by Proposition 19, we can find unit vector u21 E H21n S such that llv2 - 2121 11 < 112. Put H31 = span{ull, 211,1121,212, v4, . . . ,v , ) l ~ and let u3l E H31n S be a unit vector such that llv3 - ~ ~ < 112. ~ 1 After ) repeating this step for n times we obtain an orthonormal family {upl 1 1 5 p 5 n) such that llvp - up1II < 112 and ( ~ p lup) , = ( ~ p luql) , = 0 for P # 9 . We proceed by induction. Suppose that the finite sequences of unit vectors {up!, up2, . . . ,upk), 1 I pI n, have been constructed in S such that llvp - 'llpjll < 1/23 and (upi, uqj) = (upi,vq) = 0 for all 1 5 i, j 5 k when p # q. Let

and let u1 k + l be a unit vector chosen from HI k + l 19, such that llvl - u1 k+lll < 1 / 2 ~ + l .Next, put

n S, in virtue of Proposition

Since v2 E Hz k + l , there exists a unit vector u2 k + l E H2 k + l r l S such that 1 1 ~ 2u2 k+lll < 112"'. After repeating this for n times, we obtain the orthonormal

88

David Buhagiar,

Emmanuel Chetcuti and Anatolij Dvurehnskij

family {up k + l l 1 5 p 5 n ) in S such that (up k + l , tiqj) = (up k + l , vq) = 0 for all 1 5 j 5 k + 1 when p # q and llvp - zip k + l l l < 1/2~+'. Thus, the required sequences can be constructed by induction. PROPOSITION 21. Let {x, y} be an orthogonal pair in M E F(S) such that x E 3 and y E

m.

S.

Then there exists

Proof. By Proposition 20, there exist sequences {xi I i E N} and {yi I i E N} in S such that limi xi = X, limi yi = y and satisfy x i l y j , s l y i and y l x i for all i,j E N. Then M = {yi I i E M}" E F(S) satisfies

With these results in hand, it becomes trivial to show that: THEOREM 22. An inner product space S is complete if, and only if, F(S) = E(S). THEOREM 23. If F(S) is weakly atomic orthomodular, then F(S) = E(S). Proof. Supposing it is not the case, let M E F ( S ) \ E(S). Then there exists x E S \ (M @ M I S ) ; [XI is an atom in F(S). Since x 6 M , we have [XI A M = (0) in F(S). By hypothesis we have

(([x] V M) A MI") V M = [x] V M = [XI

+ M,

where for the last equality we simply refer to Proposition 6. Since [x]+M # M , we cannot have (([x]vM)AM'~) = {O}, and therefore there exists y E ( [ X ] V M ) A M ~ ~ and y # 0. We then have

and hence y = a x + m E M I " for some m E M. This implies x M @ M I s unless a = 0. The two conclusions lead to a contradiction and thus E(S) = F(S).

rn We end this subsection with the following corollary which will be used in the sequel. COROLLARY 24. An inner product space S is complete if, and only if, for any orthogonally closed subspace M of S and any maximal orthonormal system {xi I i E I} of M we have M = span{xi I i E I}'-sLs. Proof. We only need to show sufficiency. Let A C B be elements of F(S) and suppose that {ai I i E I} and {cj I j E J} are maximal orthonormal systems in and ALs n B = A and ALs n B, respectively. Then A = span{ai I i E I}'-"~"

Algebraic and Measuretheoretic properties

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89

span{cj I j E J ) ~ " ~One " . can easily verify that {ai, cj I i E I, j E J) is a maximal orthonormal system in B and therefore,

=AV (ALs n B). This implies that F(S) is orthomodular and therefore S is complete by Theorem rn 23.

4.2 Splitting Subspaces Criterion and the Lattice Properties of C(S) and E (S) . Unless S is a Hilbert space, neither C(S) nor E(S) can be a complete lattice. Indeed, H. Gross and H. Keller [Gross and Keller, 19771 proved that S is complete if, and only if, E(S) is a complete lattice. This was subsequently and successively strengthened as follows: S is complete if, and only if, (A) E(S) is a a-lattice [Cattaneo and Marino, 19861; (B) E(S) is a a-orthomodular poset [Dvure~enskij,19881;

(C) E(S) is atomic weakly a-complete [ ~ t &and k Weber, 19771. Recall that a poset L is said to be atomic weakly a-complete if for any sequence {a,I n E N) of atoms in L there is an infinite subsequence {hi I i E N) of {a,( n E N) such that the supremum ViENa,, exists in L. Since the atoms of E(S) are the one dimensional subspaces of S, it follows that S is complete if, and only if, C(S) is atomic weakly a-complete. 20011 that S is complete In addition to the above, it was shown in [~vure~enskij, if, and only if, E(S) satisfies the a-Riesz interpolation property. Let us elaborate on this and give a proof to the said result. A poset (P, 5 ) is said to satisfy (i) the Riesz interpolation property (RIP) if, for every al, az, bl, b2 E P with % 5 b j for i, j = 1,2, there exists an element a E P such that % 5 a 5 bj for i,j = 1,2; (ii) the a-Riesz interpolation property (a-RIP) if, for all sequences {R I i E N), {bj I j E N) in P satisfying a.j 5 bj for all i,j E N, there exists an element a E P such that a+ 5 a 5 bj for all i,j E N. It is clear that if P is a lattice (a-lattice), then P satisfies (RIP) ((a-RIP)) . THEOREM 25. An inner product space S is complete if, and only if, E(S) satisfies (a-RIP). Proof. With the aim of using Proposition 18, let x, y E 3 satisfy x l y . By Proposition 20 there exist sequences {xi I i E N) and iyiI i E N) in S such that limi xi = x, liw yi = y and satisfy x i l y j , x l y i and y l x i for all i, j E N. for all i,j E N Let Xi = [xi] and % = [yj]Lsfor all i, j E M. Then Xi for all and therefore, by (a-RIP), there exists A E E(S) such that Xi C A

s

s

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David Buhagiar, Emmanuel Chetcuti and Anatolij Dvureeenskij

i, j E N. Thus xi E A and yj E ALs and hence, x E then follows from Proposition 18.

3 and y E ALs. The result rn

It is further noted in [~vure~enskij, 20011 that the proof of Theorem 25 can be used to prove the following criterion. THEOREM 26. An inner product space S is complete i;f, and only .I;f, for every two sequences {x, I n E N) and {y, I n E N) of vectors in S such that [x,] [ym]Ls for all n , m E N, there exists M E E(S) satisfying [x,] C M C [ym]Ls for all n , m E N. This result was further strengthened in [ ~ t and ~ kWeber, 20041 by proving that an inner product space S is complete if, and only if, E(S) satisfies the atomistic subsequential interpolation property. One can recall that an orthomodular poset P has the atomistic subsequential interpolation property if for any two sequences of atoms in P, {a,I n E N) and {b, I n E N), such that a, 5 bkp for all m, n E N, there exist infinite subsequences {a,, ( k E N) and {b,, I k E N) such that one can find an element a E P satisfying a,, 5 a 5 b k for each k E N. A natural question arising from the above results is whether E(S) can be a k Weber, 20011, lattice for S incomplete. This question was answered in [ ~ t h and by a careful analysis of the lattice properties of E(S) and C(S). An inner product space S was constructed such that S is a hyperplane of and E(S) = P(S), and thus E(S) (= C(S)) is a modular lattice. However, in [Gross and Keller, 19771, it was shown that for any inner product space having a countable (infinite) linear dimension, E(S) is not a lattice. Thus, the lattice properties of E(S) do not seem to have an explicit bearing on the metric completeness of S. Let us elaborate on this. We begin by showing that for any inner product space S possessing a countable linear basis, the orthomodular poset E(S) is not a lattice. Moreover, for such an inner product space, C(S) consists of all finite and cofinite dimensional subspaces of S and therefore C(S) is a modular lattice.

+

PROPOSITION 27. Let S = [f] lf C 12,where f = CiEN (see examples in Section 2). Then E(S) is not a lattice whereas C(S) consists of all finite and cofinite dimensional subspaces of S and therefore C(S) is a modular lattice.

Proof. Let Fl = span{e2, es, es, . . . ) and Fz = span{el, es, es, . . . ). It can be easily seen that Fl @ F2= lf and that F ": = F2and F$" = Fl. The rest can be established in a series of statements. CLAIM 28.

F2

+ [f]$ E(S).

CLAIM 29. Let

+ 2)e2,+2 1 n E N, n is odd) + 2)ezn + (2n)e2,+2 I n E N, n is odd).

A =span{(2n)ezn - (2n

B =span{(2n Then

Algebraic and Measure-theoretic properties

...

(i) f E ALs , (ii) A C BLs and moreover, A 63 B = Fl, (iii) A 63 ([f]

+ B + Fz) = S.

CLAIM 30. Let

C = span((2n + 2)e2,+2 - (2n + 4)e2,+4 I n E N,n is odd) D =[ez]

+ span((2n + 4)e2,+2 + (2n + 2)e2,+4

I n E N,n is odd).

Then again (i) f E CIS, (ii) C C DI" and moreover C 63 D = Fl, (iii) C63 ( [ f

+ D + Fz) = S.

CLAIM 31. B n D

=

(0).

CLAIM 32. Let K KnL=F2+[f].

=

[f] D

+ + F 2 and L = [f]+ B + Fz. Then K , L E E(S) and

Let us give a proof to this statement. Suppose that z E K n L. Then z = d x = b y, where x, y E F2 [f]. Then d - b = y - x E F2 [f]. From the definition of B, D and F2we immediately see that d - b = 0 and therefore b = d. Since B n D = (0) we have d = b = 0 and hence z E F2 [f]. Surely we have that F2 [f] K n L and therefore K n L = F2 [f]. In the light of Claim 1 above and Proposition 13 we conclude that K A L does not exist in E(S) and therefore E(S) is not a lattice. And finally,

+

+

+

+

+

+

+

CLAIM 33. C(S) consists of all finite and cofinite dimensional subspaces of S . In view of the fact that the linear dimension of a Hilbert space is either finite or uncountably infinite, the complete subspaces of S must be finite dimensional and hence C(S) consists of all the finite and cofinite dimensional subspaces of S . H Since any two inner product spaces Sl and S2with a countable linear dimension are isomorphic (as inner product spaces), and therefore E(S1) and E(S2) are isomorphic as orthomodular posets, it follows that E(S) is not a lattice for any inner product space S with a countable linear dimension. Moreover, C(S) consists of all finite dimensional and cofinite dimensional subspaces of S and therefore C(S) is a modular lattice. We next show that E(S) = C(S) for any linear hyperplane S in 3. PROPOSITION 34. If S is an inner product space which is a linear hyperplane in 3,then E (S) = C(S).

David Buhagiar, Emmanuel Chetcuti and Anatolij Dvurehnskij

92

Proof. Let A E E(S) \ C(S). We show that if such a subspace exists, then d i m ( z / ~ )2 2. We haveS = A @ ALs, where neither A nor ALs is complete. Let a E 2 \ A and a' E ALs \ ALs. We show that {a + S,a' + S) is a linearly independent set in 3 1 s . Indeed, if cr(a+S)+p(af+S) = S, then aa+pa1 = s E S. This implies that a a + pa' = sl + s2 where 31 E A and s 2 E ALs, and therefore rn we have a a = sl and pa' = s z which is a contradiction unless a = P = 0. It is shown in [ ~ t i and k Weber, 20011 that for an infinite dimensional Hilbert space there is a dense hyperplane S in H such that E(S) (= C(S)) is not a lattice. On the other hand, in the same paper the authors show that in any separable Hilbert space H there is a dense linear hyperplane S such that E(S) is a modular lattice. We shall need the following lemma. LEMMA 35. Let H be a separable Hilbert space. Then (1) Card(H) = 2"0 . (2) Card{{vi)

I sequence in H ) = 2"o .

(3) Card{A

HI A is a complete infinite dimensional subspace of H )

= 2"o

.

THEOREM 36. Let H be a separable Hilbert space. Then H possesses a dense hyperplane S such that E(S) consists of finite or cofinite dimensional subspaces of S. Thus, E(S) is a modular lattice.

Proof. By Lemma 35 we know that the collection

IU = {A

H I A is a complete infinite dimensional subspace of H),

has cardinality 2"o. By the well-ordering principle, we can well-order the family IU. Let w be the initial ordinal number of cardinality 2"o. Then IU = { A , I a < w). We will construct a linearly independent family of vectors in H, {v, I a < w), such that for each a < w we have v, E A,. We proceed by transfinite induction. Let vl E A1 be an arbitrary non-zero vector. Suppose that the collection {up I P < a) has been constructed. Since the linear dimension of A, is 2"0 and since Card{@I P < a) < 2"0, there must be a vector v, E A, such that the family {up I P 5 a) is linearly independent. Thus we can construct the family V = {up I P < w) of linearly independent vectors of H with up E Ap. Extend this to a Hamel basis U for H. We can well-order the unbounded interval [I,co) in R to get {pa I a < w). Then define the linear map f on H into R by f (v,) = p, for all v, E V and f (u) = 0 for all u E U \ V. Then f is an unbounded linear functional on H and S = N(f) is a dense hyperplane in H. By construction, S does not contain a complete subspace of infinite dimension. rn

4.3 Quasi-Splitting Subspaces Criterion and the Lattice Properties of E4 (S)' The family Eq(S) was introduced and investigated in [Buhagiar and Chetcuti, to appear] as an intermediate between E(S) and F(S). We recall that whereas F(S)

Algebraic and Measure-theoretic properties ...

93

is always amply endowed with Boolean a-subalgebras, E(S) need not contain any Boolean a-subalgebra at all. The members of Eq(S) should behave somewhat similar to the splitting subspaces, however Eq(S) (at least when S is assumed to be separable) is always furnished with a "good supply" of Boolean a-subalgebras. Indeed, by the Gram-Schmidt orthonormalization procedure, we can convert any countable dense subset of S into an orthonormal basis {xi I i E N). Clearly, the image of the map

I' : ( I E 2")

t ,

( S n span{xi I i E I) E Eq(S))

is a Boolean a-subalgebra of Eq(S). Let us begin by giving a characterization of Eq(S) for an inner product space S.

37. Let Ml and Nl be orthogonal subspaces of 3 satisfying M1 @ PROPOSITION Nl = S. If Mo and No are subspaces of S such that Mo G MI and No 2 Nl, then Mo @ N o i s dense in S .if, and only if, Mo and No are dense in Ml and Nl, respectively. Consequently, for any closed subspace M of S , M E Eq(S) if, and only if, -Ig M I S = M (= ~ ~ 3 ) . Proof. Suppose that Mo @Nois dense in S and let us assume (for contradiction) that Mo is not dense in MI. Then we can find a nonzero vector z E M1 such that (z, x) = 0 for all x E Mo. Since z E MI, and M I I N l , we have that z is orthogonal with any vector in Mo @No,which implies that z E SIT= (0). The case when No is not dense in Nl is exactly the same and the converse is trivial. is dense in S if and only if MI" is So, for any subspace M of S, M @ MI" -IF and thus a closed subspace M of S is a quasi-splitting subspace if dense in M - -17 and only if MLs = M . rn Observe that the two inner product spaces considered in Examples 2 and 3 both have a countable linear dimension. Since, as already mentioned above, any two such inner product spaces are isomorphic (as inner product spaces), we have that E(So) g Eq(So) F(So) for any inner product space So with a countable linear dimension. We have already shown in Theorem 11that Eq(S) is orthomodular if, and only if, it is equal to E(S). Let us now investigate the lattice properties of Eq(S). It turns out that the lattice structure of Eq(S)is strongly dependent on the particular choice of the inner product space itself; Eq(S) can sometimes be a complete lattice (without S being complete) and yet, in other examples Eq(S) is not a lattice. This is in contrast with both E(S) and F(S). THEOREM 38. If S be a dense hyperplane of a Hilbert space H , then Eq(S) = F ( S ) and hence, Eq(S) i s a complete lattice.

Proof. Suppose that Eq(S) F(S) and let M E F ( S ) be an orthogonally closed subspace which is not quasi-splitting in S. Thus, there exists 0 # x E H such

David Buhagiar, Emmanuel Chetcuti and Anatolij Dvureknskij

94

that x l M and XIMI". Since M is not in Eq(S) then both M and MI"are \ M . Observe that (x, y) = 0. Since S is a hyperplane of incomplete. Let y E H, there exist nonzero scalars a and ,B such that a x ,By E S. Then a x ,By E MI"'-" = M and therefore a x = (ax + py) - ,By E M.This contradicts the fact that x l M . Hence, Eq(S) = F(S).

+

+

In Proposition 34 it is shown that if S is an inner product space which is a linear hyperplane in 3,then E(S) = C(S). Thus, in this case Eq(S) is much larger than E(S). However, as shown in [~uhagiarand Chetcuti, to appear], Eq(S) need not be a lattice. In fact we have the following result. THEOREM 39. I f S has a countable linear dimension, then Eq(S) i s n o t a lattice. So the size of Eq(S) fluctuates drastically; can Eq(S) ever coincide with E(S) for an incomplete inner product space? The authors of [Buhagiar and Chetcuti, to appear] cite the following problem. PROBLEM 40. If Eq(S) = E(S), does it follows that S is complete? We end this subsection with a partial result to this problem. THEOREM 41. Let S be a n inner product space such that d i m s / ~< co. T h e n Eq(S) = E(S) if, and only if, S i s complete. Proof. Let {yo, yl, . . . ,ynPl) be orthonormal vectors in 3\ S such that

Let K = span{yl, yz, . . . ,Y,-~}. Since K I T n S is dense in KIT (by Proposition , exists h E K L n~S such that (h, yo) # 0. Let x = yo 19) and yo E K ~ Fthere h Then x E span{yo, yl, . . . ,y , - ~ ) ~ aand yo - x = h,vo By Proposition 20 we can construct sequences {xi I i E N) and {yki I i E N) (where k = 0,1, . . . ,n- 1) in S such that limi xi = x, limi yki = yk for each k = 0,1, . . . ,n - 1 and (xi,ykj) = ( ~ k i~, k ' j )= 0 for each i,j E N and k # kt. Let us show that M = {xi I i E N ) I "is ~aSquasi-splitting subspace of S. Suppose that this is not the case, i.e. M $ Eq(S). Now yk E MLs and yki E MIS for each k = 0,1,. . . ,n - 1 and i E N. There exists a unit vector z E 3 such that and z E =lS. Since d i m S / ~= n we can find scalars a,h,. . . ,,Bn-1 such that a # 0 and w = a z + ,Boyo+ . + ,Bn-lyn-l is in S. Then w E MI" and since yk E MLs for each k, we get z E MLs which contradicts the fact that z E *l". Now suppose that M is a splitting subspace of S. Then h = hl + hz where and yo E hl E M and hz E M I s . But h = (h, yo)yo- (h, yo)x where x E which implies that hl = -(h, yo)%and hz = (h, yo)yo which is a contradiction rn since x, yo $ S .

h.

m.

-

m,

COROLLARY 42. Let S be a n inner product space such that dirn;T/S = n < co. T h e n Eq(S) is orthomodular if, and only if, S i s complete.

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4.4 Closed Subspaces Criterion. It has already been shown that for an inner product space S, the family W(S) of closed subspaces of S, forms a complete lattice when ordered by set inclusion. Moreover, we have already seen that

V Mi

=span(u

Mi)

and

/\ Mi = nMi

for any family {Mi I i E I}C W(S). The completeness criterion of S determined by W(S) is dictated by the modularity property of the lattice [Holland, 19691. Let us start by showing that unlike the other families, W(S) is not complemented in general. PROPOSITION 43. An inner product space S is complete if, and only if, W(S) is complemented. Proof. If S is a Hilbert space, then every M E W(S) satisfies M @ M I s = S and thus I is a complementation on W(S). For the converse, suppose that S is not complete. Let x E 3 \ S and M = [ x ] l n~ S E W(S). By Proposition 19 we have M = [xlL3. Thus M I S = (0) and M V M'-s = M # S. Following the work of S. Holland [1969], we inquire modular and dual-modular pairs in W(S). We recall that the pair (a, 6) where a and b are elements of some lattice L is a modular pair (expressed as (a, b)M in L) when (cVa) A b = c V (aA 6) for all c 5 6. We say that (a, b) is a dual-modular pair (and write (a, b)M* in L) when

(c A a) V b = c A (a V 6) for all c 2 6. The dual-modular pairs of W(S) can be characterized as follows. The proof of Proposition 44 relies on the fact that for any closed subspace M of S and any vector x of S, M [XI is also closed in S (Proposition 6).

+

PROPOSITION 44. Let M and N be two closed subspaces of an inner product space S. Then (M, N)M* in W(S) if, and only if, M N is closed in S.

+

To characterize the modular pairs of W(S) we need to introduce the notion of completely disjoint subspaces. Two subspaces M and N of W(S) are said to be completely disjoint if there exists a > 0 such that for all x E M , IIxII = 1 and y E N we have Ilx - yll 2 a. LEMMA 45. Let S be an inner product space and let M, N be closed subspaces of S. The following conditions are equivalent:

(1) M and N are completely disjoint;

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David Buhagiar, Emmanuel Chetcuti and Anatolij Dvureknskij

(2) There exists a

> 0 such that Ilxll + llyll 5 a11x + yll for

all x E M, y E N;

(3) M n N = (0) and the (well-defined) linear operators P(X+Y) = x, T(x+y) = y are both bounded o n the subspace M N .

+

PROPOSITION 46. If the closed subspaces M , N of a Hilbert space H satisfy M n N = (0) then M and N are completely disjoint if, and only .if, M N is closed.

+

Proof. Suppose that M and N are completely disjoint. Then there exists a such that for all x E M, y E N , we have

Let (xi

>0

+ yi) be a Cauchy sequence in M + N. Then

which implies that {xi) and {yi) are Cauchy sequences in M and N respectively. Therefore, there exist x E M and y E N such that limi xi = x and l i w yi = y. For the converse, suppose that M N is complete. Then the operators P and T defined by P ( x + y ) = x and T(x + y) = y are closed operators and therefore, by the Closed Graph Theorem, are bounded.

+

Note that the proof of Proposition 46 relies very strongly on the assumption that H is a Hilbert space. In fact, as we will see later on, the statement proved above holds only for complete inner product spaces. In the following proposition modularity is characterized in terms of complete disjointness. PROPOSITION 47. Let M and N be closed subspaces of a n inner product space S such that M n N = (0). Then (M, N ) M in W(S) if) and only if, M and N are completely disjoint subspaces of S. As a consequence of Propositions 44, 46 and 47 we have that when M , N are closed subspaces of a Hilbert space such that M n N = (0) then the pair (M, N ) is modular if, and only if, it is dual-modular. Using the orthomodular property of the projection logic of a Hilbert space it is possible to do without the assumption that M n N = (0) and hence deduce the following result. COROLLARY 48. Let M and N be closed subspaces of a Hilbert space H . T h e n (M, N ) M if, and only if, (M, N)Mf in the lattice of closed subspaces of H . In general, modularity and dual-modularity in the lattice W(S) are not equivalent. In fact, the main result of this subsection states that modularity is equivalent to dual-modularity in W(S) if, and only if, S is a Hilbert space. We shall prove the following lemma before stating the main result. LEMMA 49. An inner product space S is complete zf, and only if, M @ N is a closed subspace of S for any M, N E W ( S )such that M C N'-".

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Proof. We only need to show that the condition is sufficient for the completeness of S. Let x be a unit vector in S\ S. There exists h E S such that ( h , x ) = 1. Put y = x - h. Note that y E 3 \ S and (y,x ) = 0. By Proposition 20 there exist sequences {xi I i E N) and {yi I i E N) in S such that limi xi = x, limi yi = y and satisfy x i I y j , x i l y and y i l x for all i,j E M. Let M = s p a n s i n and N = M I s . Each of the vectors zi = xi - yi is in M @ N and limi zi = x - y = h E S. By hypothesis M @ N is closed in W ( S ) and therefore h E M @ N , i.e. h = m n where m E M and n E N. But h = x y, where x E M and y E Then x - m = - ( y - n ) , which implies that x = m and y = n and therefore x E S which is the required contradiction.

+

x.

+

THEOREM 50. An inner product space S is complete if, and only if, every modular pair is dual modular in W ( S ) .

Proof. Let M and N be any two orthogonal subspaces of W ( S ) . It is easy to check that M and N are completely disjoint. By Proposition 47 it follows that ( M ,N)M in W ( S )and hence (by hypothesis) ( M ,N)M* in W ( S ) .By Proposition 44 it follows that M @ N E W ( S ) .Result then follows from Lemma 49. 5 MEASURES ON P ( S ) , C ( S ) ,E ( S ) ,E,(S), F ( S ) AND W ( S ) If S is an incomplete inner product space, then the structure of closed subspaces resolves itself into different substructures. The aim of the rest of this survey is to study the measure-theoretic properties of these structures. A charge on any of the respective families introduced in Section 1is an additive real-valued function. Formally, if we let L be any of P ( S ) ,C ( S ) ,E ( S ) ,Eq(S)or F ( S ) ,a charge m on C is a map m : L + R, such that

m(AV B ) = m ( A )+ m ( B ) , whenever A, B E C and A I B .

(1)

We denote by Q(L)the topological linear space of all charges on C endowed with the topology of pointwise convergence on the elements of L. A charge m is said to be:

completely-additive, if equation (1) holds for any collection {Ai I i E I ) of pairwise orthogonal subspaces in C such that the supremum VierAi exists in L. Let R,,(C) be the subspace of all completely-additive charges on L. a-additive, if equation (1) holds for any sequence {Ai I i E N) of pairwise orthogonal subspaces in C such that the supremum Vie, Ai exists in C. Let Q,(C) be the subspace of all o-additive charges on C.

> 0 such that Im(A)I 5 k for each A in C; regular, if for every A in C and every E > 0 there exists a finite dimensional subspace M , contained in A, such that Im(A) - m(M)I 5 E . Let Q,(L) be bounded, if there exists k

the subspace of all regular charges on L.

David Buhagiar, Emmanuel Chetcuti and Anatolij DvureEenskij

free or singular, if m(A) is zero for every finite dimensional subspace A in L. Let !2,(C) be the subspace of all free charges on L. A measure is a positive charge on C and a state s on L is a normalized (i.e. s(S) = 1) positive charge. The set of all states on C, denoted by S(C), is a convex subset of the cube [O,1IL. When endowed with the product topology, [O, 11' is a compact space, and since S(L) is closed in [O, I]', it follows that the state space of any of the respective families described above is a compact, convex topological space. 6 GLEASON AND DOROFEEV-SHERSTNEV THEOREMS The entire theory discussed in this section depends heavily on the classical result of A. M. Gleason leason on, 19571. By solving the problem that was originally posed ~ , A. M. Gleason did not only succeed in describing by G. Mackey [ ~ a c k e19631, all the a-additive states on L(H), where H is a separable Hilbert space - he has also revealed the very delicate interplay that exists between the measuretheoretic properties of L(H) and the geometric structure of H. Gleason Theorem for bounded a-additive charges was independently proved by A. N. Sherstnev [~herstnev,19741 and A. DvureEenskij [A. DvureEenskij, 19781, and extended for bounded completely-additive charges on non-separable Hilbert spaces by M. Eilers and E. Horst [1975], T. Drisch (19791 and S. Maeda [1980] (see also [~vure~enskij, 19921). THEOREM 51 (Gleason Theorem). Let H be a Hilbert space of dimension greater than 2. For any bounded completely-additive charge m on L(H) there exists a unique bounded self-adjoint operator T on H of trace class such that

REMARK 52. Observe that in the formulation of Gleason Theorem, we can r e place 'completely-additive' with 'a-additive' if, and only if, the dimension of H is 19861). a non-measurable2 cardinal (see [~vure~enskij, Observe that Gleason Theorem shows that every bounded completely-additive charge on L(H) can be extended to a (normal) linear functional on the algebra of bounded operators on H, here denoted by B(H). We recall that B(H) is the dual space of the Banach algebra consisting of all the trace class operators endowed with the trace-norm. This duality is determined by (T, A) = tr TA, where T is a trace class operator and A E B(H). That is to say, Gleason Theorem is in fact a theorem of non-commutative integration. It can be said that Gleason 2We recall that a cardinal a is said to be non-measurable if for any set X with cardinality a, there is no 0-additive probability measure on 2 X which vanishes identically on each singleton of X. For example, No and 2 N are ~ non-measurable cardinals.

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Theorem is one of the most far-reaching mathematical results of the 20th Century. Apart from the extensive impact on the mathematical foundations of quantum mechanics, Gleason Theorem provides a solid ground for non-commutative integration theory. The basis of the proof of Gleason Theorem lies in the treatment of the real three-dimensional Hilbert space. The original proof is rather complex. Many mathematicians and physicists tried to simplify the original proof of Gleason Theorem and to extend it to more general lattices and measures (for example, measures taking values in a Banach space). For a complete documentation and citation we refer the reader to the monographs [Dvure~enskij,19921 and [Hamhalter, 20031 that treat Gleason Theorem in full detail. The most elementary proof of this Theorem is due to R. Cooke, M. Keane and W. Moran [Cooke et al., 19851, which was first presented in 1984 at a conference in Cologne, Germany. It is worth noting that if we eliminate the assumption of completeadditivity (or a-additivity when H is separable), the technique developed by A. M. Gleason cannot be used. That is why Gleason Theorem for charges on the projection lattice of a Hilbert space has been established much later by E. Christensen [christensen, 19821 for positive charges and by L. J. Bunce and J. D. M. Wright [Bunce and Wright, 19941 for the general case. (For basic definitions and terminology regarding von Neumann algebras we refer to [ ~ a d i i o nand Ringrose, 1983; Sokr, 19791 whereas for a complete treatment of the Generalized Gleason Theorem we refer to [Hamhalter, 20031.) THEOREM 53 (Generalized Gleason Theorem). Let M be a von Neumann algebra with no direct summand of Type 12. Then each bounded charge on P ( M ) extends to a bounded linear functional on M. One of the main and widely used consequences of Gleason Theorem can be easily exhibited here. DEFINITION 54. Let S(H) be the unit sphere of a Hilbert space H . A function f : S(H) + [O, 11 is called a frame-function on H if (1) f(Xx) = f (x) for all x E S ( H ) and X E D,(2) there is a constant W(f) such that

for any orthonormal basis {xi I i E I) of H. It is clear that the completely-additive charges on L(H) are in one-to-one correspondence with the frame-functions on H and therefore, in view of Gleason Theorem, for every frame function f on S(H), there exists a unique bounded trace class operator T on H, such that f (u) = (Tu, u), for all u E S(H). This means that

100

David Buhagiar, Emmanuel Chetcuti and Anatolij Dvuretenskij

which implies that f is uniformly continuous on S(H). As an immediate conse quence, observe that L(H) (dim H 3) does not admit any two-valued states. (This is related to the dispute on hidden variables in quantum theory [ ~ o c h e n and Specker, 19671.) We remark that Gleason Theorem does not hold for the case when H is two-dimensional - it is straightforward to check that L(R2) admits plenty of two-valued states. The Hahn-Jordon Decomposition Theorem is mathematical folklore in classical measure theory [Halmos, 1988, Section 291. Consequently, every completelyadditive (finite) measure on a sigma-field C is bounded. This is not the case if we replace C by the lattice of projections of a finite dimensional Hilbert space H. (See [Dvure~enskij,1992, Proposition 3.2.41.) However, S.V. Dorofeev and A. N. Sherstnev proved the deep and surprising result that if the Hilbert space is infinite dimensional then every completely-additive charge on L(H) is bounded [~orofeevand Sherstnev, 19901. We shall need the following notion.

>

+

DEFINITION 55. Let S be an inner product space. A mapping f : S(S) -+ R is said to be a frame-type function on S if: (1) for any orthonormal system {xi I i E I}in S, the series CiEI f (xi) converges; (2) for any finite dimensional subspace K of S, f 1 B(K) is a frame function on K. A frame-type function f on S is bounded if 1f (u)I 5 k for some k > 0 for all u E B(S). We can now present the remarkable result of S. V. Dorofeev and A. N. Sherstnev. THEOREM 56 (Dorofeef-Sherstnev). Any frame-type function on an infinite dimensional inner product space S is bounded. Clearly a frame function on a Hilbert space H is a frametype function. From Theorem 56 it follows that any completely additive charge m on L(H) (where dim H = co)is bounded on the atoms. By restricting m to L(N), where N is a three-dimensional subspace of H, we get a bounded charge m 1 L(N) on L(N). By Gleason Theorem, there exists a bounded symmetric bilinear form t~ on N x N such that m([x]) = tN(x, x) holds for all x E B(N) . Since every symmetric bilinear form is uniquely determined by its quadratic form, we can define a bilinear form t on H x H as follows: for any x, y E H , let N be any threedimensional subspace containing x and y, then put t(x, y) = tN(x,y). It is clear that t(x,x) = m([x]) for all x E S(H). Since m is bounded on the atoms, t is also bounded and therefore there exists a unique bounded self-adjoint operator T on H such that m([x]) = (Tx, x) for all x E B(H). The complete additivity of m entails that T is a trace class operator and m(M) = tr TPM for any M E L(H). Since T can be decomposed as the difference of two positive trace class operators, it follows that m can be expressed as the difference of two positive charges and thereby m is bounded.

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COROLLARY 57. Let H be a n infinite dimensional Hilbert space. Every completelyadditive charge o n L(H) i s bounded. In view of Corollary 57, the requirement of boundedness in the formulation of Gleason Theorem becomes superfluous for infinite dimensional Hilbert spaces. THEOREM 58. (Gleason-Dorofeev-Sherstnev) Let H be a n infinite dimensional Hilbert space and let m be a completely-additive charge o n L(H). T h e n there exists a unique bounded self-adjoint operator T o n H of trace class such that

7 IS EVERY REGULAR CHARGE ON L(H) COMPLETELY-ADDITIVE? We now investigate the relation between R,,(L(H)) and n,(L(H)). Of course, n,,(L(H)) S2,(L(H)) and if H is finite dimensional, then S2(L(H)) = O,,(L(H)) = S2,(L(H)). This is not the case when dim H is infinite. To exhibit a regular charge on L(H) which is not completely-additive we make use of the following theorem. (For the proof the reader can refer to [Chetcuti and DvureEenskij, 2003a, Theorem 3.81 and [ ~ h e t c u tand i DvureEenskij, 2004b, Theorem 3.11.) THEOREM 59. Let H be a n infinite dimensional Hilbert space. ( 1 ) Let m be a bounded charge o n L(H). Denote by X and p the infimum, and supremum of {m(A) I A E L(H)}, respectively. Then, Range(m) i s a convex set containing (A, p) .

(2) L e t s be a regular state o n L(H). For every infinite dimensional closed subspace A of H , w e have [O, 4 4 ) ) C {s(M) I M C A, dim M

< m ) C [O, s(A)].

REMARK 60. (1) Observe that in particular, if s is a state on L(H) and dimH = m , then Range(s) = [O, 11 (see [Dvure~enskijand Pt&k, 20021). If H is finite dimensional, the range of a state s on L(H) is not necessarily convex. Indeed, if H is a Hiibert space with dimension n < m , the state s(M) = dim(M)/n, M E L(H), is a state on L(H) with a discrete range. This is the only such state. (2) It is worth recalling that the range of an unbounded charge on L(H) need not be necessarily convex (see the charge constructed in Theorem 61). (3) Whereas part (1) of Theorem 59 remains valid if we replace L(H) with F(S), it fails in the case of E(S). Indeed, let S be an inner product space such that 0 < d i m s / ~= n < m. Then, the mapping s on E(S) defined by

David

Buhagiar, Emmanuel Chetcuti and Anatolij Dvureknskij

d i m p / ~ s ( M )= 7 , is a state taking at most n+ 1 values and vanishing on each complete subspace of S . (We note that it would be interesting to describe all states on E ( S ) when 0 < d i i s / ~< oo in a Gleason's type formula, see [ ~ h e t c u tand i DvureEenskij, 2004b, Exer. 8.51.)

THEOREM 61. Let H be a n infinite dimensional Hilbert space. Then, nc,(L(H)) i s a proper subset of n , ( L ( H ) ) . Proof. Let 93 = { x , I s E C) be a Hamel basis in R over the field of rational numbers. We can assume that x , > 0 for each s E C. Fix an element x,, E 93. Then every real number x E W can be uniquely expressed in the form

where a is a finite subset of C \ { s o ) and all p's are rational numbers. We define a ,!3,xs) = Pso discontinuous function 4 : W -+ Q by letting 4 ( x ) = ~ ( ~ , , x , , CsEa whenever x E W is expressed in the form of equation ( 2 ) . Let s be any regular state on L ( H ) . We show that 4 o s is a regular charge. Let E > 0 and A E L ( H ) be given. If 4 ( s ( A ) ) = 0, we take M = {0), which CsEu PSx,, where yields 14(s(A))- 4(s(M))I < E. So let 0 # s ( A ) = P,,xs0 p,, # 0. There is an integer n 2 1 such that l / n < E and x,,/n < s ( A ) . Then 0 < (p,, - l/n)x,, CsEa Psxs < s ( A ) . By (1) we can find a finite dimensional subspace M of A such that s ( M ) = (p,, -l/n)x,, +CsEu ,B8x,. Hence, \ @ ( s ( A )) 4(s(M))I= l / n < E which proves that 4 0 s is a regular charge on L ( H ) . However the charge 4 0 s is not bounded since by Theorem 59 the range of a bounded charge on L ( H ) is a convex subset of W and Range(4 o s) Q. Therefore, by Corollary 57, q5 o s is not completely-additive.

+

+

+

We conclude by giving a sufficient condition for a regular charge on L ( H ) to be completely-additive. First we prove the following elementary lemma. We prove it for E ( S ) as we will need to refer to it later on. LEMMA 62.

(1) Let m be a bounded charge o n E ( S ) . For any M E E ( S ) and 6 > 0 there exists a finite dimensional subspace Mo of M such that Im(N)I < E for every finite dimensional subspace N C M ~ M . (2) Let ml and mz be bounded regular charges o n E ( S ) such that m l ( A ) = m z ( A ) for every finite dimensional subspace A of S . T h e n ml = ma. Proof. (1) If the statement is false, then a sequence {Ni I i E N) of pairwise orthogonal finite dimensional subspaces of M can be found with the property Im(Ni)1 2 E (for each i ). This would lead to a contradiction in view of the fact that m is bounded and that one of the sets { i E N I m ( N i ) 2 E) and { i E N I m ( N i ) 5 - E ) is infinite.

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(2) Fix an arbitrary M E E ( S ) . By part (1) we can find finite dimensional subspaces Ml and M2 of M such that for any finite dimensional subspace A of M

A G M,IM =+- m l ( A ) < 614, and ACM ~ M m z ( A ) < 1214.

*

+

Let B = M1 M2. Since ml and m2 are regular it follows that Iml(MkM)l5 €14, lm2(MkM)15 ~ / 4 ,I ~ ~ ( M ; B5)€14 I and ~ m z ( M k5~ €14. ) ~ Then

Iml(M) - mz(M)I I Iml(M) - ml(B)I

+ Iml(B) - m2(B)I + Im2(B) - m2(M)I

+ Imz(M) - mz(M2) - m2(MkB)I

= I m l ( M ) - m l ( M 1 )- m l ( M f B ) I

5 l m l ( M k M ) l +I m l ( ~ f ~+)I Im 2 ( ~ ; " ) l +

Im2(~,'~>l

5 6

rn THEOREM 63. Let H be a n infinite dimensional Hilbert space and let m be a regular charge o n L ( H ) bounded o n the atoms (i-e. there exists K > 0 such that m([u]) < K for all u E B(H)). Then there exists a unique self-adjoint trace class operator T o n H such that m ( M ) = trTPM for all M E L ( H ) . Hence m i s completely-additive.

Proof. We can restrict m to finite dimensional subspaces of H and since m is bounded on the atoms we can use Gleason Theorem to obtain a bounded selfadjoint operator T on H satisfying that m ( [ x ] = ) ( T x ,x ) for all x E S ( H ) . We show that T is a trace class operator. We recall that T can be expressed as the difference of two positive operators T I and T2, and H can be split into two orthogonal subspaces H1 and Hz such that T1H2 = T2H1 = (0). Since T I is positive, to show that it is a trace class operator, it is sufficient to verify that C i E I ( T 1 x ixi) , is summable for just one orthonormal basis {xi 1 i E I ) in H . Let {xi I i E I o ) and { y j I j E Jo) be orthonormal bases of HI and H2 respectively. Then

Since m is regular, and because m is positive on all the finite dimensional subspaces of H I , it follows that m is positive (and therefore monotone) on L ( H l ) . Hence, for any finite subset I; of lo,we have 0 5 m ( [ x i ] )5 m ( H l ) . This m([xi]< ) w, and therefore T I is a trace class operator. The implies that 0 I CiEIo same can be shown for T2, and therefore it follows that T is a trace class operator. We can now apply part (2) of Lemma 62 to deduce that m ( M ) = tr T P M . H THEOREM 64 (Aarnes). L e t s be a state o n L ( H ) , where H has dimension greater than 2. Then s i s a convex combination of a completely-additive state sl and a free state s2, i.e. s = As1 (1 -X)s2.

+

David Buhagiar, Emmanuel Chetcuti and Anatolij Dvureknskij

8 MEASURETHEORETIC COMPLETENESS CRITERIA OF INNER PRODUCT SPACES The main objective of this section is the study of the interplay that exists between the linear topology of inner product spaces, ordered properties of specifk subspace structures and continuity of measures on them. It will be shown, for example, that for incomplete inner product spaces there is no hope of finding completely-additive charges on any of these subspace structures. This direction of research owes its beginning to J. Hamhalter and P. P t i k ha am halter and P t a , 19871 who proved that a separable inner product space S is complete if, and only if, S(F(S)) contains a c~-additivestate. This surprising result was after generalized for non-separable inner product spaces and completely-additive charges - see [~vure~enskij, 1992; Hamhalter, 20031 for a complete survey. Here we obtain these results as corollaries of Theorem 66. We remark that in the proof of this theorem [~uhagiarand Chetcuti, submitted], in contrast with the original proofs in [Hamhalter and Ptik, 1987; DvureEenskij, 1992; Chetcuti and DvureEenskij, 2003b], no use is made of Gleason Theorem. PROPOSITION 65. Let m be a charge on F(S). For any A, B E F(S), A C B, we have m(B) = m(A) + m(AL n B).

Proof. This follows directly from the following equalities:

As a consequence of the previous proposition observe that whenever A and B are elements of F(S) such that A C_ B and A I B = (0) then m(A) = m(B) for every charge m on F(S). THEOREM 66. Let S be an incomplete inner product space. Then, every charge o n F(S) i s free.

Proof. By Corollary 24, if S is incomplete, there exists a subspace M in F(S) and a maximal orthonormal system {xi I i E I}in M such that span{xi I i E I } ~ " ~ - " M . Let X = span{xi I i E I}lsLs. Then MIS 2 XIS. Let u be a unit vector in X I S \ M I S . Observe that u $! M @I M I s and that

s

Suppose that m is a charge on F(S) such that m([v]) # 0 for some unit vector v of S. Let U be a unitary operator on S such that U(u) = v and let ii.l be the charge on F(S) defined by k ( M ) = m(UM). Then ii.l([u]) = m([v]) # 0. By Proposition 65 we obtain ii.l(M) + fi([u]) = %(X) + ii.l([u]) = k ( X @I [u]) = k ( M which is a contradiction.

+ [u]) = fi(M), rn

Algebraic and Measure-theoretic properties . . .

105

COROLLARY 67. Let S be an inner product space. The following statements are equivalent: (1) S is complete; (2) F(S) admits a non-singular charge; (3) F(S) admits a regular non-zero charge; (4) F(S) admits a completely-additive non-zero charge.

When the Hamhalter-Ptbk Theorem was published in 1987 the problem of whether F(S) can admit a charge for an incomplete inner product space S, became eminent (see [Ptbk, 19881 and [Dvure~enskij,1992, Problem 4.3.121). Recall that if H is a Hilbert space, then 'S(L(H)) is the closure (in the topology of O(L(H))) of the convex hull of the vector states. However, the restriction of every vector state on F(S) is regular and therefore, for incomplete inner product spaces F(S) does not admit any vector state. This would somehow suggest that F(S) has no state restricting from L(H) and that F(S) could be a naturally-born example of a stateless complete lattice. Only recently it has been shown that contrary to one's intuition, the existence of a state on F(S) is not sufficient for S to be complete [Chetcuti and DvureEenskij, 2004a; Chetcuti and DvureEenskij, 2005a]. A wide class of incomplete inner product spaces for which the lattice of orthogonally closed subspaces admit a charge restricting from the projection logic of the completion was exhibited. DEFINITION 68. An inner product space S is said to be strongly dense (in its completion) if for every infinite dimensional closed subspace M of 3, we have M n s # (0). Every inner product space S satisfying d i r n s / ~< 2"0 is strongly dense (see [Chetcuti and DvureEenskij, 2005a, Proposition 21). In particular, every dense hyperplane of a Hilbert space is strongly dense. on the For every inner product space S we can define a binary relation B if there exist finite dimensional subspaces elements of F(S) as follows: A Nl, N2 of S such that A Nl = B N2. It is not difficult to check that is reflexive, symmetric and transitive, i.e. defines an equivalence relation on F(S). Let 5(S) denote the partition F(S)/-. For any A E F(S), let (A) denote the equivalence class containing A. Then ((0)) and (S) represent the classes of finite and cofinite dimensional subspaces of S, respectively. For any (A), (B) E 5(S), we say that (A) is less than (B) and write (A) 5 (B), if for every C E (A) there exists D E (B) such that C C D. Observe that (A) 5 (B) if, and only if, there exists C E (A) and D E (B) such that C C_ D. It is easily verifiable that 5 is reflexive and transitive. Moreover, if (A) 5 (B) and (B) 5 (A), then we can find subspaces C E (A) and D E (B), such that A D 5 C . Since dim CIA < oo, it follows that A D, and therefore D E (A). This implies that (A) = (B), i.e. 5 defines a partial order on 5(S). Indeed, it can be shown that 5 induces on 5(S)

+

-

-

-

+

-

-

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David Buhagiar, Emmanuel Chetcuti and Anatolij Dvureknskij

a lattice structure homomorphic to the lattice structure of F ( S ) . Moreover, for strongly dense inner product spaces we have the following theorem [Chetcuti and DvureEenskij, 2005a, Theorem 11. THEOREM 69. Let S be an inner product space such that it is strongly dense in its completion. Then 5 ( S ) is an orthomodular lattice. Furthermore, 5 ( S ) is isomorphic to (as orthomodular lattices).

5(s)

Since when S is incomplete, every charge on F ( S ) must vanish on the finite dimensional subspaces, the charges on F ( S ) are in one-to-one correspondence with the charges on Z ( S ) and hence we have the following theorem [ ~ h e t c u t i and DvureEenskij, 2005a, Theorem 21. THEOREM 70. Let S be an incomplete, strongly dense inner product space. There is a linear homeomorphism q5 : m o m @between R ( F ( S ) )and R f ( L ( H ) )(where H = 3). Each charge m on F ( S ) is the restriction of m@ in the sense that m ( M ) = m+(%), for all M E F ( S ) . The original problem (see [ ~ t & k19881 , and [~vure~enskij, 1992, problem 4.3.121) is now reduced as follows: PROBLEM 71. Does there exist an incomplete inner product space S for which F ( S ) is stateless ? Although F ( S ) does not admit any regular charge for incomplete S , E ( S ) always allows for a separating set of regular bounded charges. For any self-adjoint traceconsider the map class operator T on

s,

r n :~E ( S ) + [O, 11,

where

M

ct

tr T%.

(1)

It is clear that equation (1)defines a bounded charge on E ( S ) . We prove regularity. There exists a Let M be in E ( S ) and {yi I i E I ) be an orthonormal basis of finite subset I0 of I such that

z.

Let K E N be the cardinality of lo.By Proposition 20 there exists a finite orthonormal system {xi I i E I o ) in M such that llxi - yill < E/(~KIITII) for each

Algebraic and Measure-theoretic properties

.

i € I o Then

i.e. mT is regular. In the following theorem we show that every regular bounded charge on E(S) arises in this way. THEOREM 72. Let S be an inner product of dimension at least 3 and let m be a regular bounded charge on E(S). There exists a unique self-adjoint trace class operator T on 3 such that m(M) = trT* for all M E E(S). Proof. Restrict m to E(N), where N is any subspace of S with a finite dimension greater than 2. By Gleason Theorem, there exists a bounded Hermitian conjugatebilinear form t N on N x N such that m([x]) = tlv(x,x) for every unit vector x of N . Define a conjugatebilinear form t on S x S as follows: t(x, y) = tN(x, y), where N is any subspace of S with a finite dimension greater than 2 that contains x and y. In view of the polarization identity of Hermitian conjugatebilinear forms, it is clear that the definition of t depends only on x and y, i.e. t is well defined. Since m is bounded, t is also bounded and therefore t extends continuously to a unique Hermitian conjugate-bilinear form, again denoted by t, on 'S x 'S. Let T denote the unique self-adjoint operator on 'S such that t(x, y) = (Tx, y) for all x, y E 'S. We claim that T is a trace class operator. Let {yi I i E I ) be any orthonormal basis of 3.Let I. be a finite subset of I . By Proposition 20, for any preassigned positive number 6 we can find a finite orthonormal system {xi I i E 10) in S such that

This implies that

.

Since m is bounded, it follows that CiEI(Tyi, yi) is summable, i.e. T is a trace class operator. From part (2) of Lemma 62 it follows that m(M) = tr T*.

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David Buhagiar, Emmanuel Chetcuti and Anatolij Dvureknskij

We remark that Theorem 70 can be viewed as the generalization of Gleason Theorem for incomplete inner product spaces. It is straightforward to check that the E ~ ( 3defines ) ) an embedding (as orthomodular mapping I? : (M E E(S)) I+ posets) of E(S) into ~ ( 3 )Theorem . 70 implies that every regular bounded charge on E(S)can be lifted to a normal functional on the algebra B@) of bounded o p erators on 3 . However, it is still not known whether this holds for every bounded charge on E(S) .

(z

PROBLEM 73. Is it possible to extend every bounded charge on E(S) to a linear functional on ~ ( s ) ? In Theorem 63 we have seen that if S is a Hilbert space then, the bounded regular charges on E(S) are precisely those that are completely-additive. This is not the case when S is incomplete. Although E(S) has always an ample supply of bounded regular charges, no completely-additive charge can exist on it unless S is complete. THEOREM 74 (DvureEenskik, 1995, Theorem 3.3). A n inner product space i s complete if, and only if, there exist a frame-type function f : B(S) -+ W and a constant k > 0 such that C l f ( ~ i )Ll k iEI

for every maximal orthonormal system {xi I i E I ) in S. COROLLARY 75. A n inner product space S i s complete i f , and only if, there i s a unit vector x E S and k > 0 such that, for any maximal orthonormal system {xi I i E I ) in S, 0 < k 0 such that

and hence supiEI I fi(A)I < k. Since every operator can be decomposed as a linear combination of four positive elements, it follows that {fi 1 i E I ) is pointwise bounded on B(H) . The situation in the case that S is incomplete is still not clear - indeed the following problem is still open. Of course, when S is incomplete we replace the

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David Buhagiar, Emmanuel Chetcuti and Anatolij DvureEenskij

completely-additive charges with the bounded regular ones. Observe that in view of Theorem 72 every such charge is determined by a trace class operator. PROBLEM 78. Let S be an incomplete inner product space. Suppose that a sequence {mi I i E N) of regular bounded charges on E ( S ) converges to some charge m. Is m bounded? We recall that Nikodfm Convergence Theorem asserts that the pointwise limit of a sequence of a-additive measures {pi I i E N) on a a-field C is a a-additive measure and the set {pi I i E N) is uniformly countably additive (i.e. for any disjoint sequence { X k : k E N) in C, and for any E > 0, there exists K E N such that 1 C k > Kpi(Xk)l < E for each pi). We further remark that convergence theorems of completely-additive charges on L ( H ) , were first studied by R. Jajte [Jajte, 19721. THEOREM 79 (Nikodfm Convergece Theorem). Let H be a n infinite dimensional Hilbert space. Let {mi I i E N) be a sequence of completely-additive charges o n L ( H ) converging pointwise to some charge m E S1(L(H)). T h e n m i s completelyadditive. Moreover, the sequence {mi I i E N) i s uniformly completely-additive. Proof. Let {Mk I k E I ) be a collection of mutually orthogonal closed subspaces

of H. We can find a countable subset fi of I such that m i ( M k ) = 0 for every k E I \ Il and i E N. Each mi induces a a-additive measure on 2'1 as follows: pi : (J C I l ) mi(VkEJ Mk). Clearly, {pi 1 i E N) converges pointwise on 2'1 and hence, by the classical Nikodfm Convergence Theorem, for every E > 0 we can find a finite subset I. f Il such that

for every mi, i.e. the set { m i ( i E N) is uniformly completely-additive and therefore, the limit charge m is completely-additive. We shall now investigate the same problem for the case when S is incomplete. As we shall see, this case does not allow for such a clear answer. We say that a sequence {mi I i E N) of regular bounded charges on E ( S ) is uniformly regular if for any M E E ( S ) and for any E > 0, there exists a finite dimensional subspace Mo of M such that Imi(M) - mi(Mo)l < E for each mi. As a consequence of Theorem 63 and Theorem 79 we have the following version of the Nikodfm Convergence Theorem. COROLLARY 80. Let H be a n infinite dimensional Hilbert space. Let { m i I i E N) be a sequence of regular bounded charges o n L ( H ) converging pointwise t o some charge m E S1(L(H)). T h e n m i s bounded and regular. Moreover, the sequence {mi / i E N) i s uniformly regular. In the following theorem it is shown that the set of regular bounded charges on E ( S ) need not be sequentially closed in S1(E(S))for an incomplete S - not

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111

even if we restrict ourselves to states. For incomplete inner product spaces, the projection logic can be extremely poor and consequently the pointwise topology on n(E(S)) can be very coarse. This provides a negative answer to the problem posed by A. DvureEenskij (see [Dvure~enskij,1992, Problem 4.3.151). THEOREM 81. Let H be a n infinite dimensional, separable Hilbert space. There exists a dense hyperplane S of H such that the set of regular states on E(S) i s not sequentially closed in S(E(S)). Proof. In Theorem 36 a dense hyperplane S of H was constructed such that E(S) = P(S). Let {ei I i E N} be an orthonormal basis of S and let si : E(S) + [O, 11, M H l + i 1 12 (i E N) be the associated vector states on E(S). We claim that {si I i E N} converges (pointwise) to the unique twevalued state s in S(E(S)) assigning 0 to each complete subspace and 1 to every cocomplete subspace of S . Let u be a unit vector of S. Then limi si([u]) = liw I(u, ei)I2 = 0. This implies that limi si(M) = 0 for every finite dimensional subspace M of S, i.e. s is the unique free state in S(E(S)).

We now give a sufficient condition under which the limit of a convergent sequence of regular bounded charges on E(S) is regular and bounded. First of all, observe that the limit of a convergent, uniformly regular sequence of regular charges on E(S), is necessarily regular. The following proposition will be used in the proof of Theorem 83. It was first proved by R. Jajte [Jajte, 19721. Here we give a simpler proof without using Schur's Theorem. PROPOSITION 82 (Jajte). Let H be an infinite dimensional Hilbert space. The sequence {mi I i E N} of regular bounded charges o n L(H) converges pointwise if, and only zf, the following two conditions hold: (1) {mi([x]) I i E N} converges for each unit vector x of H; ( 2 ) for any orthonormal sequence {xk I k 6 N} in H , the series CkEnmi([xk]) converges uniformly with respect to i.

I n such case, the limit charge m i s regular and bounded, and the sequence {mi I i E N} i s uniformly regular. Proof. The necessity follows directly from Nikodfm Convergence Theorem (Theorem 79). Suppose that (1) and (2) are true. Fix Y in L(H) and let {yk I k E I} be an orthonormal basis of Y. We can find a countable subset Il of I such that for each mi, mi([yk])= 0 for all k E I \ Il. In view of condition (2), for any given positive E , there exists a finite subset loof 4, such that for all i E N, we have

In view of condition (I), let Q E N satisfy

112

for any p

David Buhagiar, Emmanuel Chetcuti and Anatolij Dvureknskij

> q 2 Q.

Then, for any p

> q 2 Q, we have

This implies that {mi I i E N) is pointwise convergent on L ( H ) . Given a positive regular charge mo on E(S), we say that the bounded regular charge m is absolutely continuous with respect to mo if, for every E > 0 there exists 6 > 0 such that Im(M)I < E, whenever M E E(S) and mo(M) < 6. Furthermore, we say that m is atomic absolutely continuous with respect mo if, for every E > 0 there exists 6 > 0 such that Im([x])l < E, whenever x E S(S) and mo([x]) < 6. A sequence {mi I i E N) of bounded regular charges on E(S) is uniformly absolutely continuous with respect to mo if, for every E > 0 there exists 6 > 0 such that Imi(M)I < E for every mi, whenever M E E ( S ) and mo(M) < 6.

THEOREM 83. Let S be a n inner product space and let {mi I i E N) be a sequence of bounded regular charges o n E(S) converging pointwise t o m. Suppose that there exists a positive regular charge mo o n E(S) such that every mi i s atomic absolutely continuous with respect to mo. T h e n m i s bounded and regular and the sequence {mi ( i E N) i s uniformly regular. Proof. Denote by Ti (i = 0,1, . . . ) the corresponding bounded self-adjoint trace for all M E E(S). class operator on 3 associated with mi, i.e. mi(M) = trTj* First we prove that the sequence {('I'z, z) I i E N) converges for each unit vector z of 3. From the following inequalities

it follows that llTill 5 (lTollfor all i = 1,2,. . . . Fix a unit vector z in 3 and let 6 > 0 be arbitrary. There exists a unit vector x of S such that 11% -211 < (llTo11c)/6. In addition, we are guaranteed that for some I E N, I((Tp- Tq)x,x)I < €13 for all p > q 2 I. Thus, for p > q 2 I, we have

and therefore {(Tiz, z) I i E N) is convergent.

Algebraic and Measure-theoretic properties .. .

Let {zk I k E N) be an arbitrary orthonormal system in (TOzk,zk)l < 00,there exists a positive integer K such that

113

S.

Since

CkEN 1

for any 6 > 0 and i E N. If we denote by riii the the charge on E(S)defined by fii(N) = tr TipN,for N E E(S),then it is clear that the sequence {riii I i E N) satisfies conditions (i) and (ii) of Proposition 82. This implies that {riii I i E N) converges pointwise on E(S) to some bounded regular charge r72 on ~ ( 3 ) It . is clear that m is the restriction of f i , i.e. m(M) = r72(M) for all M E E(S). The uniform regularity of {mi I i E N) follows from that of {riii I i E N). Indeed, for any M E E(S) and E > 0 there exists a finite dimensional subspace Mo of % such that Iriii(M) - riii(Mo)l < E for each i. Using Proposition 20 a finite dimensional subspace N of M can be found with the property that Imi(M) - mi(N)( 5 6 for rn each i. Let C be a a-algebra of subsets of a set X and denote by ca(C) the linear space of all a-additive measures on C endowed with the topology of pointwise convergence. Suppose that p is a positive a-additive measure on C. Then, the Vitali-Hahn-Saks Theorem asserts that if K ca(C) is a relatively compact set such that each element of K is absolutely continuous with respect to p on C, then K is uniformly absolutely continuous with respect to p (see [ ~ u n f o r dand Schwartz, 19581). In [chetcuti and Hamhalter, to appear, Example 4.51 it was shown that this cannot be carried over directly to the non-commutative case. In fact, the Vitali-Hahn-Saks Theorem fails for L(H) when dim H is infinite. This demonstrates a different character of non-commutative measure theory as far as charges are concerned. However, in [Chetcuti and Hamhalter, to appear, Theorem 4.61 it is proved that if we restrict to positive charges then the Vitali-Hahn-Saks Theorem holds for L(H). We thus have the following theorem. THEOREM 84 (Vitali-Hahn-Saks Theorem). Let H be a n infinite dimensional Hilbert space and let {mi I i E N) be a pointwise convergent sequence of positive regular charges o n L(H) such that such that each mi i s absolutely continuous with respect to some regular state so. Then, {mi I i E N) is uniformly absolutely continuous with respect to so. We conclude this section by exposing another contrast between L(H) and E(S) when S is incomplete. The Vitali-Hahn-Saks Theorem can fail in the E(S) setup even when we restrict to states. The inner product space that works out to be a counter-example for this is the same one considered in Theorem 36, i.e. E(S) = P(S). Let {siI i E N) be the sequence of vector states corresponding to the orthonormal basis {ei I i E N) of S . Put s = $si. Observe that each si is absolutely continuous with respect to so and {si ( i E N) converges pointwise to the state assigning 0 to finite dimensional subspaces and 1 to the co-finite dimensional ones. However, it is not difficult to check that {siI i E N) is not uniformly absolutely continuous with respect to so.

xiEN

David Buhagiar, Emmanuel Chetcuti and Anatolij DvureEenskij

10 INFINITEVALUED MEASURES Here we present an overview of some of the results concerning measures on the projection lattice of a Hilbert space that can take also infinite values. An example of such a measure would be the dimension function. A mapping p : L(H) + [0,m] is said to be a measure if p(M @ N) = p(M) p(N) for all M, N E L(H) such that M I N . As we did with charges in Section 5, we can distinguish between the notions of (1) completely-additive, (2) a-additive and (3) regular measures. Observe that here by a regular measure we mean that given E > 0 and M E L(H), there is a finite dimensional subspace N of M such that p ( ~ Nn ~ H

LEMMA 85. Let Hn be an n-dimensional Hilbert space, where n 3. Let p be a measure o n L(Hn) such that p(Hn) = m and suppose that P i s a n ( n - 1)dimensional subspace such that p(P) < m. If q is a unit vector such that p([q]) < m , then q E P.

Proof. First assume that Hn = It3. Suppose that q is a unit vector making an acute angle 0 < a with the plane P such that p([q]) < m . We can choose a coordinate system as follows. Let XI be the unit vector bisecting the angle a. Let x2 be a unit vector orthogonal to X I and q. Observe that x2 E P . Let x3 be a

Algebraic and Measure-theoretic properties

. ..

115

unit vector orthogonal to xl and x2. Since 2 2 is orthogonal to q, it follows that the plane containing x2 and q has a finite measure. Denote this plane by Q. Put

F = {x E t5(iR3) 1 3

q') E P x Q such that pllq' and x E span{pl, q')).

Using the geometry of the sphere, one can show that F contains all the unit vectors with the absolute value of the latitude (with respect to the xlxz-plane) less than or equal to a. Hence we can find a unit vector ql and a plane PI such that p([ql]) < co, PI) < oo and the angle between ql and PI is 2a. One can repeat the same argument and deduce the p([x]) < oo for all x in B(W3) having absolute value of latitude equal to 2a. Therefore, we can repeat the above inductively and conclude that 4W3) < co, which is a contradiction. Now, we consider the case when H, = (C3. Let yo be a unit vector perpendicular to P and yl, y2 be two orthogonal vectors in P such that (q, yi) is real for i = 0,1,2. Let M be the threedimensional completely-real subspace generated by {yo,y1, y2}. Then q E M. The measure p induces a measure fi on L ( M ) by letting P([x]) = p([x]) for every x E B(M). Observe that fi([yo]) fi([y~]) fi([y2]) = oo and therefore, from the above arguments, it follows that q = Xlyl X2y2, where Xi E W, i.e. q E P . To prove it for every n > 3, one can apply induction. Suppose result is true for n < k and let P be a (k - 1)-dimensional subspace of Hk such that p(P) < oo. Let q be a unit vector with p([q]) < oo, and u a unit vector in P orthogonal to q. Put Hk-1 = [uILHkand P' = [u]IHkn P. Observe that q E Hk-1 and dim P' = k - 2. By induction hypothesis it follows that q E P'.

+

+

+

If t is a symmetric (not necessarily bounded) bilinear form and P is a projection on H such that P H D(t) then, we can define the symmetric bilinear form t o P on H x D(t) as follows: t 0 P(x, y) = t(Px, y). We write t o P E Tr(H) if t o P is bounded, and the associated bounded self-adjoint operator is a trace class operator. Moreover, we denote by tr t o P the trace of this operator, i.e.

where {xi I i E I}is any orthonormal basis of H . Using Lemma 85, it is possible to prove the following theorem. THEOREM 86 (Lugovaja and Sherstnev, 1980). Let H be a Hilbert space of dimension greater than 2. Let p be a n-finite, n-additive measure. The following assertions are equivalent: (1) There exists a (unique) positive, symmetric bilinear form t with the domain D(t) dense in H such that trtoPM iftoPMETr(H), otherwise.

David Buhagiar, Emmanuel Chetcuti and Anatolij Dvurehnskij

(2) p has a support, i.e. a closed subspace M, of H such that p(N) = 0 if, and only if, N IM,. (3) p is a completely-additive measure. EXAMPLE 87. Any positive self-adjoint operator T defines by equation (1)a afinite and completely-additivemeasure, where t is defined by t(x, y) = (T'/'x, ~ ' 1 ' ~ ) with D(t) = D(T'/~). We now exhibit a difficulty that arises when we use bilinear forms that are not determined by positive self-adjoint operators. EXAMPLE 88. Let H be an infinite dimensional separable Hilbert space. Let {en I n E N) be an orthonormal basis, which is a part of a Hamel basis {ei I i E I)of H . Fix an element ei, which does not belong to the orthonormal basis {en I n E M) and define a linear operator T : H 4 H and a linear functional f : H 4 cC via

and

where cri, is the scalar corresponding to eb in the decomposition x = CiEI ai ei, x E H, with respect to the given Hamel basis. If we set t(x, y) = (Tx, Ty), then t defines via equation (1) a a-finite measure that is not a-additive.

G.D. Lugovaja [1983] showed that on a separable Hilbert space, a singular positive symmetric bilinear form does not define a a-finite, a-additive measure by equation (1). For that reason the following important result was proved by B. Simon [1978]: THEOREM 89. Let t be a densely defined positive symmetric bilinear form on a Hilbert space H. Then there exist two positive symmetric bilinear forms t, and t, with D(t,) = D(t) = D(t,) such that

where t, is the largest closable bilinear form less than t. THEOREM 90 (Lugovaja, 1983). Let H be an infinite dimensional Hilbert space and let t be a singular, positive symmetric bilinear form with dense domain D(t). Then, the mapping p : L(H) 4 [0,ca]defined by equation (1) is not a a-additive measure on L(H). The affirmative answer is also due to G. D. Lugovaja:

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117

THEOREM 91 (Lugovaja, 1983). Let H be an infinite dimensional Hilbert space. A positive symmetric bilinear form t with a dense domain D ( t ) defines through equation (1) a a-additive measure o n L(H) if, and only if,

(m), E Tr(H)

implies t o PM€ Tr(H),

for any M E L(H), where ( t o P M ) i~s the regular part of the closure t o PM. THEOREM 92. Let H be a separable Hilbert space of dimension greater than 2. Let p be a a-finite, regular measure o n L(H). Then, there exists a unique positive bilinear form t with a dense domain such that equation ( 1 ) holds. Conversely, for any positive bilinear form t o n H such that D ( t ) is dense in H, the mapping pt : L(H) + [0, m ] given by trtoP~

iftoPM€Tr(H), otherwise

i s a a-finite, regular measure. We end the survey by quoting another problem. PROBLEM 93. Characterize the a-finite, a-additive signed-measures taking infinite values on L(H) for a separable infinite dimensional Hilbert space H, where by a signed measure we mean a mapping p : L(H) [-m, m] preserving additivity on orthogonal subspaces. (It is possible to show that every signed measure attains at most one of the values k m . ) ACKNOWLEDGEMENTS The third author was supported by the Center of Excellence SAS - Physics of Information - 1/2/2005, the grant VEGA No. 216088126 SAV and by Science and Technology Assistance Agency under the contract No. APVT-51-032002. Bratislava, Slovakia. BIBLIOGRAPHY [Aarnes, 19701 J. Aarnes, Quasi-states on C*-algebras, Tl-ans. Amer. Math. Soc. 1 4 9 (1970), 601-625. [Amemip and Araki, 1966471 I. Amemiya and H. Araki, A remark on Piron's paper, Publ. Res. Inst. Math. Sci., Kyoto Univ. Ser. A 2 (1966-67), 423-427. [Bennett and Foulis, 19971 M. K. Bennett and D. J. Foulis, Interval and scale effect algebras, Adv. Appl. Math. 9 1 (1997), 823-834. [Birkhoff and von Neumann, 19361 G. Birkhoff and J. von Neumann, The logic of quantum mechanics, Ann. Math. 37 (1936), 823-843. [Bunce and Wright, 19941 L. J. Bunce and J. D. M. Wright, The Mackey-Gleason problem for vector measures on projections in von Neumann algebras, J . London Math. Soc. (2) 4 9 (1994), 133-149. [Buhagiar and Chetcuti, 20041 D. Buhagiar and E. Chetcuti, On isomorphism of inner product spaces, Math. Slovaca 5 4 (2004), 109-117.

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[Buhagiar and Chetcuti, t o appear] D. Buhagiar and E. Chetcuti, Quasi-splitting subspaces i n a prehilbert space, Math. Nach. ( t o appear). [Buhagiar and Chetcuti, submitted] D. Buhagiar and E. Chetcuti, Only 'freeJmeasures are admissible o n F ( S ) when S is incomplete, submitted. [Cattaneoand Marino, 19861 G. Cattaneo and G. Marino, Completeness ofinner product spaces with respect to splitting subspaces, Letters Math. Phys. 11 (1986), 1Ft20. [Chetcuit, 20021 E. Chetcuti, Completeness Criteria for Inner Product Spaces, MSc thesis, Univ. o f Malta, 2002. [Chetcuti and DvureEenskij, 2003al E. Chetcuti and A . Dvureknskij, Range of charges on orthogonally closed subspaces of an inner product space, Inter. J . Theor. Phys. 42 (2003), 19271942. [Chetcuti and DvureEenskij, 2003bl E. Chetcuti and A. DvureEenskij, A finitely additive state criterion for the completeness of inner product spaces, Letters Math. Phys. 64 (2003), 221227. [Chetcuti and DvureEenskij, 2004al E. Chetcuti and A . DvureEenskij, The aistence of finitely additive states o n orthogonally closed subspaces of incomplete inner product spaces, Letters Math. Phys. 67 (2004), 75-80. [Chetcuti and DvureEenskij, 2004bl E. Chetcuti and A. DvureEenskij, Measures on the splitting subspaces of an inner product space, Inter. J. Theor. Phys. 43 (2004), 364-384. [Chetcuti and DvureEenskij, 2005al E. Chetcuti and A. DvureEenskij, The state-space of the lattice of orthogonally closed subspaces, Glasgow Math. J . 47 (2005), 213-220. [Chetcuti and DvureEenskij, 2005bl E. Chetcuti and A. Dvureknskij, Recent progress o n preHilbert-space logics and their nzeasure spaces, Inter. J . Theor. Phys. 44 (2005), 2139-2151. [Chetcuti et al., 20061 E. Chetcuti, P. de Lucia and A. Dvureknskij, Sequential convergence of regular measures o n pre-Hilbert space logics, J . Math. Anal. Appl. 318 (2006), 194-210. [Chetcuti and Hamhalter, t o appear] E. Chetcuti and J . Hamhalter, Vitali-Hahn-Saks Theorem for vector nzeasures o n operator algebras, Oxford Quarterly J . Math. ( t o appear). [Chevalier et al., 20001 G. Chevalier, A. DvureEenskij and K . Svcrtil, Piron's and Bell's geometric lemmas and Gleason's theorem, Found. Phys. 30 (2000), 1737-1755. [Cooke, 19781 T . A. Cooke, The Nikodym-Hahn-Vitali-Sakstheorem for states o n a quantum logic, "Mathematical Foundations o f Quantum Theory" A. R . Marlow ed., Acad. Press, London (1978), 275-285. [Christensen, 19821 E. Christensen, Measures o n projections and physical states, Comm. Math. Phys. 86 (1982), 529-538. [Cooke et al., 19851 R . Cooke, M. Keane and W. Moran, A n elementary proof of Gleason's theorem, Proc. Cambr. Phil. Soc. 98 (1985), 117-128. [de Lucia and Pap, 20021 P. d e Lucia and E. Pap, Convergence theorems for set functions. In: Pap, E. (ed.), Handbook o f Measure Theory. Vol. I Amsterdam, North-Holland, 2002 pp. 125-178 . [Dorofeevand Sherstnev, 19901 S.V. Dorofeev and A.N. Sherstnev, Frame-type functions and their applications, Izv. vuzov matem. no. 4 (1990), 23-29 ( i n Russian). [Drish, 19791 T . Drisch, Generalization of Gleason's theorem, Inter. J . Theor. Phys. 18 (1979), 239-243. [Dunford and Schwartz, 19581 N. Dunford and J. T . Schwartz, Linear Operators, Vol. 1, Interscience, New York, 1958. [ A . DvureEenskij, 19781 A. Dvureknskij, Signed states on a logic, Math. Slovaca 28 (1978), 33-40. [Dvureknskij, 19861 A. Dvureknskij, Generalizations of Maeda's thwrem, Inter. J . Theor. Phys. 25 (1986), 1117-1124. [ ~ v u r e k n s k i j19881 , A. DvureEenskij, Completeness of inner product spaces and quantum logic of splitting subspaces, Letters Math. Phys. 15 (1988), 231-235. [Dvureknskij, 19921 A. DvureEenskij, Gleason's Theorem and Its Applications, Kluwer Acad. Publ., Dordrecht, Ister Science Press, Bratislava, 1992. [Dvureknskij, 19931 A. DvureEenskij, Gleason's t h w r e m and completeness criteria, Inter. J . Theor. Phys. 32 (1993), 2377-2388. [Dvureknskij, 19951 A. DvureEenskij, Frame functions and conzpleteness of inner product spaces, Ann. Inst. Henri PoincarB, 62 (1995), 429-438. [ ~ v u r e k n s k i j20011 , A. Dvureknskij, A new algebraic criterion for completeness of inner product space, Letters Math. Phys. 58 (2001), 205-208.

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[Dvureknskij, 20031 A. Dvureknskij, States on subspaces of inner product spaces with the Gleason property, Inter. J. Theor. Phys. 42 (2003), 1393-1401. [ D v u r e n s k i j and Pulmannovi, 19891 A. DvureEenskij and S. Pulmannovi, A signed measure completeness criterion, Letters Math. Phys. 17 (1989),253-261. [Dvureknskij and Pulmannovi, 20001 A. DvureEenskij and S. Pulmannovi, New 'Ii-ends i n Quantum Structures, Kluwer Acad. Publ., Dordrecht, 2000. [Dvureknskij et al., 19901 A. DvureEenskij, T. Neubrunn and S. Pulmannovi, Finitely additive states and completeness of inner product spaces, Found. Phys. 20 (1990), 1091-1102. [Dvureknskij and P t i k , 20021 A. DvureEenskij and P. P t i k , O n states on orthogonally closed subspaces of an inner product space, Lett. Math. Phys. 62 (2002),63-70. [Eilers and Horst, 19751 M. Eiers and E. Horst, The theorem of Gleason for nonseparable Hilbert space, Inter. J. Theor. Phys. 13 (1975),419-424. [Giuntini and Greuling, 19891 R. Giuntini and H. Greuling, Toward a formal language for unsharp properties, Found. Phys. 19 (1989), 931-945. [Gleason, 19571 A. M. Gleason, Measures on the closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885-893. [Gudder, 19751 S. P. Gudder, Inner product spaces, Amer. Math. Monthly 82 (1975), 251-252. [Gudderbook, 19881 S. P. Gudder, Quantuna Probability, Academic Press Inc., Boston, San Diego, New York, Berkely, Tokyo, Toronto, 1988. [Gross and Keller, 19771 H. Gross and H. A. Keller, O n the definition of Hilbert space, Manuscr. Math. 23 (1977), 67-90. [Halmos, 19881 P. R. Halmos, Measure Theory, Springer-Verlag, New York, Heidelberg, Berlin, 1988. [Hamel, 19051 G. Hamel, Eine Basis allen Zahlen und die unstetigen Losungen der Funktionalgleichung: f (x Y ) = f (x) f(Y), Math. Ann. 60 (1905),459-462. [Hamhalter, 20031 J. Hamhalter, Quantum Measure Theory, Kluwer Acad. Publ., Dordrecht, 2003. [Hamhalter and Pt&, 19871 J. Hamhalter and P. P t i k , A completeness criterion for inner product spaces, Bull. London Math. Soc. 19 (1987), 259-263. [Harding, 19911 J. Harding, Orthonaodular lattices whose MacNeille conapletions are not orthomodular, Order 8 (1991), 93-103. [Harding, 19961 J. Harding, Decompositions i n quantum logic, mans. Amer. Math. Soc. 348 (1996), 1839-1862. [Holland, 19691 S. Holland, Jr., Partial solution to Mackey's problem about modular pairs and completeness, Canad. J. Math. 21 (1969), 1518-1525. [Holland, 19951 S. Holland, Jr., Orthomodularity in infinite dimensions; a theorem of M. SolCr, Bull. Amer. Math. Soc. 32 (1995),20S234. [Husimi, 19371 K. Husimi, Studies on the foundations of quantum mechanics 1, Proc. Phys.math. Soc. Japan 19 (1937), 766-789. [Jajte, 19721 R. Jajte, O n convergence of Gleason measures, Bull. Acad. Polon. Sci., ser. Math. Astr. Phys. 20 (1972), 211-214. [Kadison and Ringrose, 19831 R. V. Kadison and J. R. Ringrose, Fundamentals of the Theory of Operator Algebras, Vols I and 11, Academic Press, London, 1983 and 1986. [Kalmbach, 19831 G. Kalmbach, Orthonaodular Lattices, Acad. Press, London, New York, 1983. [Kalmbach, 1986) G. Kalmbach, Measures and Hilbert Lattices, World Sci. Publ. Co., Singapore, 1986. [Keller, 19801 H. A. Keller, Ein nicht-klassischer Hilbertischen Raum, Math. 2. 172 (1980), 41-49. [ ~ o c h e and n Specker, 19671 S. Kochen and E. R. Specker, The problem of hidden variables i n quantum mechanics, J . Math. Mech. 17 (1967), 59-67. [Kolmogorov, 19331 A. N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung,Berlin, 1933. [KBpka and Chovanec, 19941 F. KBpka and F. Chovanec, D-posets, Math. Slovaca 4 4 (1994), 2134. [Lugovaja, 19831 G. D. Lugovaja, Bilinear forms defining measures on projectors, Im. vuwv rnatem. No. 2 (1983),88-88 (in Russian). [Lugovaja and Sherstnev, 19801 G. D. Lugovaja and A. N. Sherstnev, O n the Gleason theorem for unbounded measures, h v . vuzov matem. No. 2 (1980), 3&32 (in Russian).

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[Mackey, 19631 G. W . Mackey, Mathematical Foundations of Quantuna Mechanics, Benjamin, New Y o r k , 1963. [Maeda, 19801 S. Maeda, Lattice Theory and Quantuna Logic, Maki-Shoten, Tokyo, 1980 (in Japanese). [Maeda and Maeda, 19701 F. Maeda and S. Maeda, Theory of Symmetric Lattices, SpringerVerlag, Berlin, 1970. [Mayet, 19981 R. Mayet, Some characterizations of the underlying division ring of a Hilbert lattice by automorphisms, Inter. J . Theor. Phys. 37 (1998), 109-114. [Murray and von Neumann, 19361 F. J . Murray and J. von Neumann, O n rings of operators, Ann. Math. 37 (1936), 116-229. [Murray and von Neumann, 19371 F. J . Murray and J . v o n Neumann, O n rings of opemtors ZZ, 'Ikans. Amer. Math. Soc. 4 1 (1937), 208-248. [Murray and v o n Neumann, 19431 F. J . Murray and J . v o n Neumann, O n rings of opemtors ZV, Ann. Math. 44 (1943), 7164308. iron, 19641 C. Piron, Asionaatique quantique, Helv. Phys. Acta 37 (1964), 439468. [Piron, 19761 PirIPir C . Piron, Foundations of Quantum Physics, Addison-Wesley, Reading, Mass., 1976. [Piziak, 19901 R. Piziak, Lattice theory, quadratic spaces, and quantuna proposition systems, Found. Phys. 20 (1990), 651-665. [PtAk, 19881 P. P t i k , FAT - C A T ( i n the state space of quantum logics), Proceedings o f "Winter School o f Measure Theory", Liptovski J i n 1988, (Czechoslovakia) pp. 113-118. [ ~ t and & Pulmannovi, 199l] P. PtAk and S. Pulmannovi, Orthonaodular Structures as Quant u m Logics, Kluwer Acad. Publ., Dordrecht, 1991. [ P t i k and Weber, 19771 P. Ptdk and H. Weber, Two remarks on the subspace poset of an inner product space, Contrib. Gen. Alg. 10, Proceedings o f t h e Klagenfurt Conference, May 19-June 1, 1997, Verlag Johannes Heyn, Klagenfurt, 1998, pp. 263-267. [PtAk and Weber, 20011 P. P t i k and H. Weber, Lattice properties of subspace fanailies of inner product spaces, Proc. Amer. Math. Soc. 129 (2001), 2111-2117. [ ~ t and & Weber, 20041 P. PtAk and H. Weber, Order properties of splitting subspaces i n an inner product space, Math. Slovaca 54 (2004), No. 2, pp. 119-126. [PulmannovB, t o appear] S. Pulmannovi, A note on honaomorphisms of inner product spaces, Demonstratio Math., t o appear. [Sherstnev, 19741 A. N. Sherstnev, O n a notion of a charge i n nonconanautative scheme of measure theory, Veroj. metod i kibern. Kazan, no. 10-11 (1974), 68-72. (in Russian). [ ~ i m o n19781 , B. Simon, A canonical decomposition for quandmtic fornls with application t o monotone convergence theorems, J . Funct. Anal. 28 (1978) 377485. [Solkr, 19951 M. P. Sol&r, Characterization of Hilbert spaces by orhtomodular spaces, Comm. Algebra 23 (1995), 219-243. [Solbr, 19791 M. P. Solkr. Theory of Operator Algebras I, Springer, Berlin-Heidelberg-New Y o r k , 1979. [Varadarajan, 19851 V . S. Varadarajan, Geometry of Quantuna Theory, Springer-Verlag, New York Inc., 1985. [von Neumann, 19291 J . v o n Neumann, Zur Algebra dur Funktionaloperationen und T h w r i e der norn~alenOperatoren, Math. Ann. 102 (1929), 370-427. [von Neumann, 19401 J . von Neumann, O n rings of operators ZZZ, Ann. Math. 4 1 (1940), 94161. [von Neumann, 19491 J. von Neumann, O n rings of operators. Reduction theory, Ann. Math. 50 (1949), 401485. [Wigner, 19591 E. P. Wigner, Group Theory and its Applications t o Quantum Mechanics of Atomic Spectra, Acad. Press. Inc. New Y o r k , 1959.

HANDBOOK O F QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM STRUCTURES Edited by K. Engesser, D. M. Gabbay and D. Lehmann O 2007 Elsevier B.V. All rights reserved

121

QUANTUM PROBABILITY Stan Gudder

1 INTRODUCTION

This chapter begins with the more concrete and familiar and then gradually progresses to the more general and abstract. To save space and spare the reader from too many technical details, we have decided to omit some of the longer proofs. However, we provide references for the proofs of all stated results and encourage the reader to further pursue topics of interest. Some proofs are provided for recent results to give the reader a flavor of the techniques that are now employed. It is widely believed that quantum theory is intrinsically statistical. That is, the study of elementary particles and their interactions is irreducibly probabilistic and cannot be derived from an underlying deterministic theory. Even those who advocate that such underlying hidden variable theories may be possible admit that they do not seem to be practical and do not appear to offer anything new or useful. In any case, just making the assumption that quantum theory is probabilistic goes a long way toward establishing the structure of quantum mechanics and understanding its interpretations. An important component of any physical theory is the concept of measurement. When a measurement is performed, the physical system interacts with a measuring apparatus and in general the environment cannot be neglected. The environment produces noise, any real apparatus has limited resolution and efficiency and the experimental control of initial conditions is never exact. It follows that no real measurement is absolutely precise and results of measurements must be described statistically. Hence we have a second reason for the importance of probability in quantum theory. Quantum probability generalizes classical probability theory in two ways. First, a traditional random variable f describes a measurement that is absolutely precise. After all, the values f (w) are sharply defined with no fuzziness involved. The uncertainty of classical probability comes from our lack of knowledge of the situation. We can only make statistical predictions about which sample point w occurs. However, as we have previously mentioned, real measurements are never completely precise. There are always errors involved and these should be taken into account in a realistic theory. Second, in a quantum mechanical world, two different measurements A and B frequently interfere with each other. One way this is manifested is that they

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cannot be performed simultaneously. We must specify which measurement is performed first and which is performed next. This results in a sequential or temporal product A o B of the measurements. Due to interference, in general, we have A o B # B o A so the theory becomes noncommutative. For this reason, quantum probability is sometimes called noncommutative probability theory. Classical probability theory contains no mechanism for describing such situations. Until recently the main framework for such descriptions has been operators on a Hilbert space which provides an obvious noncommutativity. In this case, fuzzy quantum events are represented by operators satisfying 0 5 A 5 I and observables or measurements are represented by positive operator-valued measures. These will be considered in Sections 4-8. In the last few years, researchers in the foundations of quantum mechanics have developed a more general formalism called a sequential effect algebra. In this case, fuzzy quantum events are represented by abstract objects with certain properties called effects and observables are represented by a-morphisms from the Borel subsets of the real line into the set of effects (effectvalued measures). These will be discussed in Section 9.

2 MOTIVATIONAL CALCULATIONS This section presents some standard concrete calculations that begin to show how the basic structure of quantum mechanics follows from probabilistic considerations and furthermore how quantum probability differs from classical probability theory. Suppose we measure the coordinate s of a quantum particle. Assuming that quantum theory is probabilistic, this measurement should be described by a random variable Q, on the probability space (B3,p) representing the position of the particle in three-space W3 where p(r, t) is a probability density for r E B3 depending on the time parameter t. The probability that Q, has a value in the Borel set A E B(B3) at time t is given by

where dV = dxdydz and the expectation of Q, at time t is r

So far, this is just classical probability theory. However, one of the first basic axioms of quantum mechanics is the Born rule which states that p(r, t) = I$(r, t)12 where $(r, t ) is a complex-valued function called a probability amplitude. We ~ , for conclude that $(r, t) is a unit vector in the complex Hilbert space L ~ ( RdV) each fixed t. We can write the probability amplitude as $ = f i e i s l where sl(r,t) is a measurable real-valued function. We can also define the measurable real-valued function s2(r,t) by s 2 = -(1/2)lnp so that fl = e-'=. Defining the complexvalued function s = sl is2 we obtain the eikonal form = eis. The function s 2

+

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is directly related to the probability density p and is easily computed. However, the function 31, which is peculiar to quantum mechanics, is completely arbitrary unless the theory is developed further. We now show that the Heisenberg uncertainty principle essentially follows from the Born rule. We assume that p, s1,s2 are differentiable functions. Since p is a probability density, integrating by parts gives

lim px = 0 we obtain

Sx

dV = -1. Applying Schwarz's inequality and assuming that the integrals exist, we have

Under the reasonable assumption that

x+&m

To simplify the second integral on the right side of (1) we have

Since --

dx

dx

we obtain

Applying (2) and (3) we have

Substituting (4) into (1) gives

Now

(2) ( 2 )(2)

=i

so we conclude from (5) that

s2

l Za$l 2

2

= e2s2

1Ek1

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Equation (6) is equivalent to the uncertainty principle which states that

where ti is a constant determined by experiment (Planck's constant). Assuming that p is symmetric so that Et(Q,) = 0 we conclude that the variance of Q, is

l2

Interpreting J 1-ih d V as the variance of a momentum measurement Px in the x-direction, (7) gives

The uncertainty principle (8) is a statistical inequality and in the process of o b taining (8) we have derived the momentum operator -ih& using probabilistic methods. The fact that p is a function of t is a concept that is usually not present in classical probability theory and is due to the motion of the particle in space. Information about the motion of the system can be described by the function sl. Using the notation XI = x, xz = y, xs = z, we define the probability velocity vector

where m is the mass of the particle. The change of probability in time is determined by the probability density current jk = puk, k = 1,2,3. The equation for jk is analogous to the expression for the density current in fluid dynamics. We then obtain

Since

+=fi

eisl

and

+*

=

we conclude that

Since jk and p are real-valued functions, taking the real part gives

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The existence of a nonzero jk depends on the fact that the probability amplitude I) is complex-valued. This is a feature of quantum mechanics that is not present in classical probability theory. Notice that jk can be written in terms of the momentum operators 1 j k = g ($*PXkI) - +PXk$*) which further indicates a relation between jk and the motion of the particle. The previous calculations suggest that the state of a quantum system is r e p resented by a unit vector $ in a Hilbert space and that position and momentum observables in the x-coordinate are represented by the self-adjoint operators

Motivated by these concrete calculations, the next section establishes the notation and definitions of a mathematical framework for quantum probability. 3 NOTATION AND DEFINITIONS

Let H be a complex Hilbert space that represents the state space of a quantum system S. The set of effects E(H) for S is the set of operators on H satisfying 0 5 A 5 I. More precisely, A E E(H) if and only if 0 5 (Ax, x) 5 (x, x) for all x E H. Effects represent yes-no measurements that may be unsharp (imprecise, fuzzy). F'rom a probabilistic viewpoint, effects can be thought of as fuzzy quantum events. It is interesting that many of the important classes of quantum operators are given by subsets of E(H). For example, sharp yes-no measurements are represented by the set of projection operators P(H) &(H). In traditional quantum mechanics projections correspond to (sharp) quantum events. A state for S is represented by an operator W E E(H) satisfying tr(W) = 1. We call W a density operator and denote the set of density operators by D(H). The density operators correspond to the probability measures of classical probability theory. The pure states of S are given by D(H) n P(H). A more general measurement (not just twevalued) is represented by a positive operator-valued (POV) or effect-valued measure X : B(R) -+ E(H) [Busch et al., 1995; Busch et al., 1991; Davies, 19761. That is X(B) = I and X(UAi) = C X ( A i ) for Ai f~ A, = 0, i # j, and convergence of the summation is in the strong operator topology. In this case, X(A) represents the effect observed when the measurement X has a value in the Bore1 set A. A projection-valued (PV) measure X : B(R) + P ( H ) corresponds to a sharp measurement and by the spectral theorem gives a single self-adjoint operator called an observable. In this work we shall mainly consider discrete measurements which are given by a countable set {Ai} G E(H) such that C Ai = I. Then Ai corresponds to the effect observed when the measurement results in outcome i. For A E E(H), the negation of A is defined by A' = I - A. Of course, A' E &(H) and if P E P ( H ) then P' E P(H). For A, B E E(H), if A B 5 I we

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define A @ B = A B. Then A @ B is defined if and only if A 5 B'. Roughly speaking, A@B corresponds to a parallel measurement of A and B. The algebraic system (E(H), @, 0, I) forms an effect algebra. Effect algebras are important in i j PulmannovB, 2000; foundational studies of quantum mechanics [ ~ v u r e ~ e n s kand Foulis and Bennett, 1994; Giuntini and Greuling, 1989; Gudder and Greechie, 20021. An orthoposet is an algebraic system (P, 0 , l ) where (P, j b k , and thus b V bta = b V ( M A a) = a by orthomodularity, too. Also, (b ") a = (b' A a)' A a = (b V a') A a = b by Lemma 4. This proves that b H bfa is an orthocomplementation in [0, a]. Orthomodularity follows from ort homodularity in L. rn The logic of Lemma 9 will be denoted by L[O, a]. LEMMA 10. If for every a, b E L the supremum a V b exists in L then the logic L is a a-lattice.

Proof. Let (ai)i be a sequence of elements of L. Put bl = a l , . . . ,b, = a1 V a2 V . . V a, (n E U). Then bl ,b2 A b', , . . . is a sequence of orthogonal elements, hence rn ai. its supremum, b, exists. It is easily seen that b =

-

VE1

Note that a logic L becomes a Boolean a-algebra if it is a distributive a-lattice. The next theorem provides a characterization of Boolean a-algebras among logics. THEOREM 11. A logic L is a Boolean a-algebra if and only if a a, b E L.

tt

b for all

Proof. Necessity is clear. For sufficiency, if a * b for all a, b E L, then by Lemma 6, a V b and a A b exist in L for all a, b E L. By Lemma 10, L is a a-lattice. Distributivity follows from Lemma 8. A logic L which is also a lattice is called a lattice logic. Obviously, a lattice logic is an orthomodular a-lattice. Let us say that (al, a2, as) C L is a distributive triple if ai A (aj V ak) = (ai A aj) V (ai A ak) for every permutation {i, j, k) of the set {1,2,3). THEOREM 12 (Foulis-Holland). Let a, b, c be elements in a lattice logic L such that a chosen one is compatible with the other two. Then (a, b, c) is a distributive triple.

Proof. Assume, e.g., that a * b, a * c. By Lemma 8 we have a tt b V c and a ~ ( b v c= ) ( a ~ b ) V ( a ~ cIt) .suffices toprove that (aVb)Ac= (aAc)V(bAc). By the De Morgan laws this is equivalent to (a A b) V c = (a V c) A (b V c). But ( a v c ) ~ ( b ~=c( )a ~ c A) [ ( ~ A ( ~ V C ) V ~ ' A = ( ~a~ VC ( b) V ] c ) ~ c =( a ~ bV)C , where we used Lemma 8 and compatibility of aV c with a A ( b c)~and a' A (b V c).

rn A subset M of a lattice logic L which has the property that among any three arbitrarily chosen elements of M there is one which is compatible with the other two, is called a Foulis-Holland set. Greechie [Greechie, 19791 (see also [~almbach, 19831 for a simpler proof) proved that the sublattice of L generated by a FoulisHolland set M is distributive. The following lemma is a strengthening of Lemma 8 for lattice logics.

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-

THEOREM 13. Let L be a lattice logic. Let M be a subset of L such that V M := V{m : m E M ) exists in L, and let b cr m for all m E M. Then b V M and V{bAm:rn€m)=(VM)~b.

-

Proof. Evidently, (V M ) A b 2 m A b for all m E M. Let u E L be such that u > m A b f o r a l l m E M . T h e n z : = u A ( V M ) A b ~ m ~ b f o r a l l rM n .~ m for all m E M it follows that We will show that z = (V M) A b. From b bA(btV(VM)') 5 bA(b'vmt) = b A r n t < bA(VM)'. By Lemma7, b - V M . From orthomodularity we have that (V M ) A b = z V d, where d I z. Thus d 5 ( V M ) A b a n d d 5 z' 5m'Vb'. Thisentailsthat d 5 (m'Vbf)Ab=m'Abfor all m E M . Hence d 5 (V M)', which together with d 5 V M entails that d = 0. We have shown that u 2 (V M) A b, so that (V M) A b is the supremum of the set { b ~ m : m ~ M ) . DEFINITION 14. Let L1 and L2 be logics. A mapping h : L1 + L2 is said to be a a-homomorphism if (i) h(1) = 1; (ii) h(at) = h(a)' (a E L1);

VEl h(ai)

(iii) h ( V z l ai) = elements of L1.

for every sequence (ai)i of mutually orthogonal

A bijective a-homomorphism h such that h-I is also a a-homomorphism is called an isomorphism. If L1 = La, then an isomorphism is called an automorphism. A a-homomorphism is a lattice a-homomorphism if h(a V b) = h(a) V h(b) whenever a V b exists in L1. If the condition (iii) is required to be satisfied only for finite orthogonal sequences, then h is called a homomorphism. Let us observe that if h : L1 t Lz is a a-homomorphism and (ai)i is an arbitrary sequence of elements in a lattice logic L1, then h(Vi ai) = Vi h(ai). Note also that if L1 is a lattice logic and L2 is an arbitrary logic, then every a-homomorphism h : L1 + L2 is a lattice a-homomorphism exactly when L1 is a Boolean a-algebra [~afiasovb,19811. DEFINITION 15. A subset L1 of a logic L is called a sublogic when L1 is a logic with the ordering and complementation inherited from L and when the identity mapping id : L1 -+ L is an injective a-homomorphism. A sublogic is called a lattice sublogic if the following implication holds: Whenever (ai)i is a sequence of elements in L1 such that Vi ai exists in L, then Via* belongs to L1. A sublogic L1 of a logic L is called a Boolean sublogic if (i) for every sequence (ai)i of L1 the supremum Vi ai exists in L and belongs to L1; (ii) every (a, b, c) c L1 is a distributive triple. DEFINITION 16. We say that a logic L is regular if for every a, b, c in L which are pairwise compatible we have a * b V c (equivalently, a * b A c).

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Let A be an arbitrary subset o f a logic L . Put C ( A ) := {a E L : a * b f o r every b E A } . W e easily see that, for every A , B

(i) A C B (ii) A

c L,

=+ C ( B ) c C ( A ) ;

c C(C(A));

(iii) C C ( C ( A ) )= C ( C C ( A ) )= C ( A ) . T h e set C ( L ) is called the center o f L. L E M M A 17. For every subset A L , the set C ( A ) is a sublogic of L . Proof. Evidently, 0 and 1 belong t o C ( A ) . If a E C ( A ) , then also a' E C ( A ) b y Lemma 5. I f (ai)iENis a sequence o f pairwise orthogonal elements in C ( A ) , then Viewai exists in L , and for every b E A we have b * ai, i = 1 , 2 . ... Hence b A ai exists. A s ( bA ai)iEn is an orthogonal sequence, ViEN bA ai exists in L . B y Lemma rn ai, and hence ViEN ai E C ( A ) . 8, b * ViEN L E M M A 18. If B is a subset of pairwise orthogonal elements in a regular logic L , then C ( C ( B ) ) is a Boolean sublogic of L . Proof. B y Lemma 17, C ( C ( B ) )is a sublogic o f L . Since B is a pairwise compatible set, we have B C C ( B ) , which entails that C ( C ( B ) ) = C ( C ( C ( B ) ) ) ,so that C ( C ( B ) )is a compatible set. Let a , b E C ( C ( B ) )and c E C ( B ) . T h e n a , b, c are pairwise compatible. T h e n regularity implies that c * a V b, so that a V b E C ( C ( B ) ) . It follows that C ( C ( B ) )is a lattice. Moreover, ( a ,b, c) C C ( C ( B ) )is a distributive triple for every a, b, c. PROPOSITION 19. The center C ( L ) of any logic L is a Boolean sublogic of L . Proof. B y Lemma 17, C ( L ) is a sublogic o f L . I f a , b E C ( L ) , then a * b, therefore a V b exists in L . Moreover, a A c and b A c exist in L for all c E L , and in addition, a A c tt b A c. B y Lemma 8, aV b E C ( L ) . Since a * b for all a, b E C ( L ) , C ( L ) is a Boolean sublogic. THEOREM 20. Let L be a regular logic. Let K := { a , : a E I } be a subset of pairwise compatible elements in L . Then there exists a Boolean sublogic B of L such that K B C_ L . Proof. W e have K C C ( C ( K ) ) ,and b y Lemma 18, C ( C ( K ) )is a Boolean sublogic of L. rn DEFINITION 21. A logic L is called separable if every set o f nonzero pairwise orthogonal elements in L is at most countable. PROPOSITION 22. Let L be a separable lattice logic. For every set A C L there exists a countable subset (ai)iErpisuch that V A = V z 1ai. Consequently, L is a complete lattice.

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Proof. Let A c L. Let B denote the set of all suprema of at most countable subsets of A. Let C := {b A c' : b E B, c 5 b}. By Zorn's lemma, there exists a maximal subset of pairwise orthogonal elements in C , let us say { b l A d , b2Ac&,. . .}, bi. Then clearly b E B. If there which is at most countable. Let b = is a E A such that a $ b, then 0 # (b V a) A b' E C. Then the inequalities b i A d 5 b 5 (bVa)'Vbfori=1,2, ...imp lythat ( b ~ a ) ~ Ibb i' A c ' , , i = 1 , 2 , ..., which contradicts the maximality of the set { b i < ~ ) E l . This proves that b = V A.

Vzl

rn

DEFINITION 23. An element a in a logic L is called an atom if a inequality b 5 a implies either b = 0 or b = a.

# 0 and the

DEFINITION 24. A logic L is called atomic if for every b E L, b # 0, there is an atom a E L with a 5 b. A logic L is called atomistic if every element in L can be written as a supremum of all atoms that are less than or equal to L. PROPOSITION 25. A lattice logic is atomic if and only if it is atomistic. Proof. Let L be an atomic lattice logic. Let b E L, b # 0 and let A be the set of all atoms lying under b. Then A # 0. Let c E L be such that a 5 c for all a E A. Then a 5 b A c, and by orthomodularity, b = (b A c) V (b A (b' V c')). If b A (b' V c') # 0, then, by the atomicity of L, there is an atom e 5 b A (b' V c'). But e 5 b implies e 5 b A c, in contradiction with e 5 b' V c'. Hence b = b A c, and rn therefore b = V A.

Let us note in concluding this paragraph that Prop.25 does not have to hold for general logics (see [Greechie, 19691). 3 COMPATIBLE SUBSETS OF A LOGIC In the quantum logic approach to quantum mechanics, a quantum logic L is interpreted as the set of all experimentally verifiable propositions about a physical system. The compatible subsets of L should correspond to those propositions that can be simultaneously verified, and hence can be dealt with as classical propositions. Generally, we may assume that the elements of a subset M of L are compatible if they can be embedded into a Boolean sublogic of L, in accord with the assumption that the logic of a classical system is a Boolean D-algebra. In this section, we will find intrinsic characterizations of compatible subsets of a logic. If a logic L is regular, then by Theorem 20, every pairwise compatible subset of elements of L is contained in a Boolean sublogic of L. The following example shows that this need not hold in an arbitrary logic. EXAMPLE 26. Let M = {1,2,3,4,5,6,7,8) and let L be the set of all subsets of M with an even number of elements. Then L ordered by inclusion and endowed with set-theoretical orthocomplementation is a logic. Let A = {{I,2,3,4}, {1,2,5,6}, {2,3,5,7}}. It is easy to check that elements of A are

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pairwise compatible. But they cannot be contained in a Boolean sublogic of L since {1,2,3,4,5,6)) A {2,3,5,7) does not exist in L. As a result, a stronger condition of compatibility is needed to characterize those subsets of a logic that can be embedded into Boolean sublogics of L. Such a condition, so-called strong compatibility, was found independently in [GUZ, 19711 and [Neubrunn, 19741. Let us introduce it in the next two definitions. DEFINITION 27. Let M be a subset of a logic L. We say that a, b E M are M compatible in M , written a * b, if there are pairwise orthogonal elements a l , bl, c in M such that a = a1 V c and b = bl V c. DEFINITION 28. A subset A of a logic L is said t o be strongly compatible if for every two elements a, b E A, a L(A) * b, where L(A) is the smallest sublogic of L that contains A. Obviously, the smallest sublogic of L that contains A, L(A), exists (L(A) is the intersection of all sublogics of L that contain A). Recall that the smallest sublogic of L that contains A is called the sublogic of L generated by A. THEOREM 29. Let A C L, and let L(A) be the sublogic of L generated by A. Then L(A) is a Boolean sublogic of L if and only if A is strongly compatible. Proof. If L(A) is a Boolean sublogic, then A C L(A) implies that for every a, b E A, a = a A (b V b') = (a A b) V (a A b') by distributivity in L(A) . Since L(A)

is a Boolean sublogic, a A b, a A b' E L(A), hence a L(A) ++ b. Conversely, assume that A is strongly compatible. For a E A, put B, := {b E L(A) : a L(A) * b). By the properties of compatibility, B, is a sublogic of L, and since A c B, by strong compatibility, we have L(A) c B, for every a E A. Now let b E L(A), and put Bb := {c E L(A) : b LLA) c). The first part of this proof implies that A c Bb, and hence L(A) c Bb for all b E L(A). This entails that every pair a, b E L(A) is compatible in L(A). As a consequence, L(A) is a Boolean sublogic of L. Obviously, strong compatibility implies pairwise compatibility. The next example shows that strong compatibility is not equivalent to pairwise compatibility, and also that strong compatibility is not necessary for a subset of L to be embeddable into a Boolean sublogic of L. EXAMPLE 30. Let M = {1,2,3,4) and let L be the Boolean algebra of all subsets of M. Let A = {{1,2), {2,3), {3,4)). The smallest sublogic L(A) containing A consists of all subsets with an even number of elements. The elements of A are pairwise compatible in L but not strongly compatible, (e.g., {1,2) n {2,3) = (2) which does not belong to L(A)). In what follows, we will use the following notation. Let D = (-1, I), and let A C L. Write A' := {a' : a E M) and let a' := a, a-I := a'. Put D := {-1,l).

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Let A = {al, az, . . . ,a,) be a finite subset of L such that for every d E Dn the infimum a? A a? A . . . A a$ exists in L. Then put

and set

com(al,a2,. . . , a n ) =

V{f : f E FA).

DEFINITION 31. Let A = {a1, az, . . . ,a,) be a finite subset of a logic L. We say that A is a compatible set if all elements in FAexist and com(al, az, . . . , a,) = 1. If A is an arbitrary subset of L then A is said to be compatible if all finite subsets of A are compatible. The element com(a1, az, . . . ,a,) (if it exists) is called the commutator of the set {al, az, . . . ,a,). If L is a lattice, then com(A) exists for any finite subset A c L. For any two elements we have com(a, b) = (a A b) V (afA b) V (aA b') V (afA bf ) . This commutator in , (see also [chevalier, 1984; orthomodular lattices was introduced in [ ~ a r s d e n19701 Chevalier, 1989; MatouSek, 19921, etc.). THEOREM 32. A subset A of L is contained in a Boolean sublogic of L if and only if A is a compatible set. Proof. If there is a Boolean sublogic Lo of L such that A C Lo, then A is a compatible set. Assume that A is a compatible subset of L. If M = {al, az, . . . ,a,) is a finite subset of A, then FM= {a:' A a$ A .. A a$ : d E D) is a set of pairwise orthogonal elements of L the supremum of which equals 1. Let B(M) denote the set of suprema of all subsets of FM.It is easily verified that B(M) is a Boolean sublogic of L. Put B = U{B(M) : M a finite subset of A). Then B is also a Boolean sublogic of L. Indeed, if al, az, a3 E B and if MI, M2, M3 are finite subsets of A with ai E B(Mi), i = 1,2,3, then MI U Mz U M3 is a finite subset of A, and B(Ml U M2U M3) contains all a l , a2, as. It follows that any three elements of B are contained in a Boolean sublogic of L. Therefore B is a Boolean subalgebra of L. We still need to verify that B is closed under the formation of suprema of countable subsets. Let Lo be the sublogic of L generated by B. We will show that Lo is a Boolean sublogic of L. Take a E B and put K, := {b E Lo : b 3 a). By Lemma 17, K , is a sublogic of Lo. Since B is Boolean, we have B C Ka. Thus Ka = Lo for all a E B. Further, take b E Lo and put Kb := {c E Lo : c 3 b). Then, again, Kb is a sublogic of Lo. By the first part of this proof, B C Kb. It b in Lo for all a, b E Lo. follows that Kb = Lo for all b E Lo. Hence we have a By Theorem 11, Lo is Boolean a-algebra, hence Lo is a Boolean sublogic of L.

-

The following condition was introduced in [Brabec, 19791.

.

DEFINITION 33. A subset A of a logic L is f-compatible if for every finite subset F = {al, a2,. . . ,a,) of A there exists a finite set G = {g1,g2,. . . ,gk) of pairwise orthogonal elements of L such that for every i = 1,2, . . . ,n there is a subset Hi C G such that ai = V{fj : f j E Hi). The set G is called an orthogonal cover of F.

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The following statement can be proved by a technique similar to the proof of , Theorem 32 ( [ ~ t b kand PulmannovB, 1991, Prop. 1.3.221, see also [ ~ r a b e c1979; Brabec and PtBk, 19821). PROPOSITION 34. Let A be a finite subset of L. Then A is compatible if and only if it is f-compatible. By Lemma 9, for any b E L, the interval [0, b] can be viewed as a logic with the smallest element, 0, the greatest element, b, and with the relative orthocomplementation xlb = x' A b. In the next lemma, we prove that compatibility in L and compatibility in [0,b] are equivalent. LEMMA 35. Let b E L. The elements al, a2,. . . ,a, in the interval [0,b] are compatible in L if and only if they are compatible in [0,b].

Proof. If com(al, a2,. . . , a,) exists and equals to 1, the it is easy to see that com(al, as,. . . , a,) A b is the commutator of al, aa, . . . , a, taken in [0, b], and since it is equal to b, the compatibility in [0,b] follows. Conversely, if elements are compatible in [0,b], then they have an orthogonal cover, which means that they are f-compatible and hence by Proposition 34 they are compatible in L. It is easily seen that for a tweelement set A = {al, a2) the notions of pairwise compatibility, f-compatibility and compatibility coincide. We have already shown that pairwise compatibility is strictly weaker than compatibility. The next theorem and Pulmannovb, 1991, Theorem 1.3.251) shows that also "triplewise", (see [ ~ t B k etc. compatibility is too weak to guarantee compatibility. THEOREM 36. Let n (n 2 2) be a natural number. Then there exists a (finite) logic L, with a subset A, such that the following statements hold: (i) the set A, is not compatible in L,; (ii) every subset of A, which contains less that n elements is compatible in L,. Let us recall ([Harding, 1998; Harding, 1839-1862; PtSk and Pulmannovb, 19911, etc.) that a logic L is said to be regular if for any set {a, b,c) C L of pairwise compatible elements we have a tt b V c. As seen before, there are logics that fail to be regular (Example 26). Interestingly, projection logics are usually regular e r KatrnoBka, 1979; Harding, 19981 and [ P ~and B Weber, (see e.g. [ ~ l a c h s m e ~and 20011). PROPOSITION 37. Every lattice logic is regular. Proof. The proof follows immediately from Lemma 8. PROPOSITION 38. A logic L is regular if and only if every pairwise compatible subset of L admits an enlargement to a Boolean sublogic of L.

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Proof. By Theorem 20, every pairwise compatible subset of a regular logic is contained in a Boolean sublogic. To prove the converse, assume that every pairwise compatible subset of a logic L is compatible. Then if a, b, c are pairwise compatible elements of L, then they are contained in a Boolean sublogic Lo of L, hence b V c rn exist in L and belongs to Lo, and a tt b V c in Lo implies a * b V c in L. PROPOSITION 39. (i) Let A be a compatible subset of L. Among the Boolean sublogics of L containing A there is a maxzmal and a minimal one. (ii) The union of all maximal Boolean sublogics of L is L. (iii) The intersection of all mosimal Boolean sublogics of L is C(L).

Proof. Let K A denote the collection of all Boolean sublogics of L that contain A. By Theorem 32, K A is nonempty. The intersection, Lmi,, of K A is the smallest Boolean sublogic that contains A. Let us consider the collection K A ordered by inclusion. By a standard application of Zorn's lemma the collection K A conThis completes the proof of the statement. The tains a maximal element, L,,. rn statements (ii) and (iii) are obvious consequences of the statement (i). DEFINITION 40. A maximal Boolean sublogic of L is called a block of L. In view of Proposition 39 every logic can be viewed as a union of its blocks. A more detailed analysis of the configuration of blocks in a logic can be found in [Greechie, 1971; Matouavsek, 20041 and [Navara and Rogalewicz, 19911. 4 STATES ON A LOGIC DEFINITION 41. A state on a logic L is a mapping s : L -+ [0, 11, where [O,1] denotes the unit interval of the real line W,which satisfies the following conditions: (i) s(1) = 1; (ii) if (ai)i is a sequence of pairwise orthogonal elements in L, then 00 s (Vi=l ai) = ~(ai).

czl

+

If sl and sz are states on L, then their convex combination s = am1 (1 -a)sz, where 0 5 a 5 1, are also states on L. A state s on L is called pure if s = as1 (1 - a)s2, 0 < a < 1 implies that s = sl = sz. A state which is not pure is called a mixture. If for every a E L, s(a) E (0, I), we say that s is a two-valued state. Let S(L) denote the set of all states on L and let Sz(L) denote the set of all twevalued states on L. Let us note that every s E S2(L) is pure. Indeed, if we can write s = as1+ (1 - a)s2 for a E (0,l) and for distinct sl, s 2 E S(L), then there is 0 < a < 1with sl(a) # s2(a), and the value asl (a) (1-a)s2(a) is neither 0 nor 1. Moreover, if L is a Boolean logic, then s E L2(L) 8 s is pure. Indeed, if s E S(L) is such that 0 < s(b) < 1 for some b E L, we can put s = as1 (1 - a)sz, where

+

+

+

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Pave1 Ptdk, Sylvia Pulmannovd

a = s(b) and sl, s 2 are states on L defined by the formulas sl(a) = s(b)-ls(a A b) and s2(a) = (1 - s(b))-ls(a A b'). Thus s is not pure. LEMMA 42. The set S(L) is a a-convex set, that is, if sl, s 2 , . . ., are states and q 2 0, Czl q = 1, then s = C z f l s i is also a state on L.

Proof. It follows from the fact that s(a) = CEl cisi(a) 5 CEO=, ci = 1 for every rn a E L. DEFINITION 43. We say that a state s on a logic L has the Jauch-Piron property, or is a Jauch-Piron state if the following implication holds: If s(a) = 1and s(b) = 1 for some a, b E L, then there is c E L, c 5 a, c 5 b such that s(c) = 1. Equivalently, a state s is Jauch-Piron if s(a) = 0 = s(b) implies that there is d E L, a, b 5 dl and s(d) = 0. If L is a lattice logic, then a state s is Jauch-Piron iff s(a) = 1 = s(b) implies s(a A b) = 1 (a, b E L). In the study of Jauch-Piron states we encounter new phenomena in (non-commutative) measure theory see e.g. [ ~ Lucia e and Ptik, 1992; Bunce et al., 1985; Muller, 1993; Ruttimann, 1977; Ptik, 19981. DEFINITION 44. We say that a set S of states on L is (i) order determining if s(a) 5 s(b) for all s E S implies a 5 b; (ii) unital if for every a E L, a

# 0 there is s E S such that

s(a) = 1;

(iii) rich if {s E S : s(a) = 1) C {s E S : s(b) = 1) implies a 5 b. (Equivalently, if a , l b implies 3s E S : s(a) = l,s(b) # 0). We say that a logic L is (j) unital if it has a unital set of states, (jj) rich if it has a rich set of states, (jjj) rich2 if it admits a rich set of (0,l)-states. LEMMA 45. (i) A rich set S is unital and order determining. (ii) If L is a lattice logic, then a unital set M of Jauch-Piron states is rich.

Proof. Part (i) is straightforward. For (ii) Let a $ b. Then a = (a A b) V c with c # 0, c Ia A b. Let s(a) = 1 s(b) = 1. Then the Jauch-Piron property implies that s(a) = 1 =+ s(a A b) = 1. By unitality there is s E M such that s(aAb) = 1, contradicting s(c) = 1. But then s(c) = 1 =+ s(a) = 1 cIaAb. rn s(a) 5 s(b). In the As a consequence of orthomodularity we have a 5 b definition of a state, orthomodularity is not explicitly used. Consequently, the n e tion of a state makes sense for any partially ordered orthocomplemented set which

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is closed under suprema of countable pairwise orthogonal sets. Orthomodularity 19931 and or am halter and Ptbk, 19871). is often forced (see also [~vure~enskij, PROPOSITION 46. Let (L, 5,') be a partially ordered orthocomplemented set with the properties (i)-(v) of Definition 1. Let S be an order determining set of states on L. Then L also has property (vi) of Definition I , that is, L is a logic. Proof. Let a 5 b, a, b E L. Then a V b' exists in L, and s(a V b') = s(a) + s(bf) = s(a) + 1- s(b). For any s E S we have s ( a V (a v b')')

=

s(a) = s(a)

+ s((a V b')') + 1- s(a v b')

= s(b),

and since S is order determining, this implies that a V (a V b')' = b. This is equivalent to (vi) of Definition 1. The following definition introduces an important class of set-representable logics. It should be noted that even finite logics are not necessarily set-representable, see [Katrno~ka,1982; Ovchinnikov, 1997; Tkadlec, 1993; Sherstnev, 19681, etc. DEFINITION 47. Let R be a nonempty set and let A be a collection of subsets of R. Then A is said to be a concrete logic if the following conditions are satisfied:

3. if {Ai : i E N) C A is a countable family of pairwise disjoint subsets of R, then U{Ai : i E N) E A It is interesting to note that the notion of a concrete logic has long been known in classical probability theory in connection with generating Bore1 sets (see e.g. [Olej~ek,19951 for more details). There, it is called a Dynkin system of sets. Note also that there are Boolean logics (Boolean cr-algebras) which are not concrete [~tbk and Pulmannovi, 19911. The following statement was proved in udder, 19791. THEOREM 48. A logic L is isomorphic (as a logic) to a concrete logic if and only if L is rich2. Proof. Suppose first that L is isomorphic to a concrete logic (R, A). Then we may identify L with A. Take A, B E A such that A $ B. Then A \ B # 0 and therefore we can choose a point p E A \ B. Consider a state sp concentrated at p (i.e., sp(C) = 1 iff p E C). Then sp(A) = 1 and sp(B) = 0. Thus, L is rich2. Conversely, suppose that L is rich2. Put R = S2(L), and A = {A E S2(L) : there is an a E L such that A = {s E S2(L) : s(a) = 1)). We show that (0, A) is a concrete logic. If A = {s E S2(L) : s(a) = I), then R\A = {s E S2(L) : s(af) = 1). Hence A E A implies R \ A E A. Let Ai = {s E S2(L) : s(ai) = I), i = 1 , 2 , .. .,

Pave1 Pt6k, Sylvia Pulmannov6

160

.

and let Ai be mutually disjoint sets (i E N). As L is rich2, it follows that the elements a i , i E N are mutually orthogonal. We may then write Ui,,Ai = {s E S2(L) : s(ViENai) = 1). Hence (a,A) is a concrete logic.

The following proposition gives a characterization of compatibility in concrete logics. PROPOSITION 49. Let A be a concrete logic of subsets of $2. Then A, B E A are compatible if and only .if A n B E A.

Proof. Assume that A * B. Then there are mutually disjoint sets A1, B1, C in A such that A = A1 U C, B = B1 U C. Then A n B = C E A. Now assume that A n B E A. Then we have A = ( A n B ) u ( A \ ( A n B ) ) , B = ( A n B ) U ( B \ ( A n B ) , where the sets A n B, A \ ( A n B), B \ (A n B) are mutually disjoint sets. Since for disjoint elements of A the set-theoretical union U coincides with the supremum in A. It follows that A * B.

.

The state space S(L) of a logic can be rather poor (even for finite L we can have S(L) = 8 - see [Greechie, 1971; Navara, 19941 and [Weber, 19941) or rather bizarre ([Navara et al., 1988; Navara and Rogalewicz, 19881. To illustrate that, let us consider the following four classes of (unital) logics, denoted by C1, La, C3, C4:

- L E L1 e if a cft b, there is a state s E S(L) such that s(a) = 1and s(b) # 1; if a cft b, and if we are given an s E S(L) such that s(a) = 1 and s(b) 2 1- E ;

- L E C2

E

> 0,

then there is a state

- L E C3 H if a cft b, then there is a state s E S(L) such that s(a) = 1= s(b); - L E L4 e if a cft b, then there is a twevalued state s E S2(L) such that s(a) = 1= s(b) and, moreover, the set S2(L) is unital for L. THEOREM 50. [ ~ t i and k Pulmannovi, 1991, Th. 2.4.121 (i) L E C1 e L is a rich logic. (ii) L E L4 H L is a concrete logic. (iii) The inclusions C4 C C3 C L2 C L1 hold.

Proof. (i) We rephrase the definition of a rich logic: a logic L is rich if for any a, b E L with a 5 b there is a state s E S(L) such that s(a) = 1 and s(b) # 1. Thus every rich logic obviously belongs to L1. Now let L E L1 and let a, b E L, a 5 b. If a * b, we have a = a1 V c, b = bl V c, where a l , bl, c are mutually orthogonal. Obviously, a1 # 0. Take a state s E S(L) with s(al) = 1. Then we have s(a) 2 s(al) = 1. From b 5 a; we have s(a;) = 0 2 s(b). Hence s(a) = 1 and s(b) = 0. (ii) Follows by (i) and Theorem 48. (iii) The inclusions C4 C C3 C C2 follow immediately from the definitions. It remains t o verlfy that C2 c C1. Suppose that L E C2 and take elements a, b E L. bb', and therefore there is a state s E S(L) such that s(a) = 1 If a cft b, then a and s(bb')2 112. Since s(b) = 1- s(b1), we see that s(b) < 1/2, as required.

+

.

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It can be proved that all the inclusions in Theorem 48 are proper [ ~ t b k and Pulmannovb, 1991]. The counter examples are, usually, constructed with the help of so called Greechie past job [Greechie, 1971; Greechie, 19691 or its generalizations [Mayet et al., 20001. Let us present here a typical way of constructing interesting features of state spaces in this manner. Let us first introduce another class of logics ([~ogalewicz,1984b] and [ ~Btand Rogalewicz, 1983bl). We point out that various other such classes having their origin in mathematical analysis have been r Navara, 1987; D7Andreaand de Lucia, 1991; De Simone and studied [ ~ i n d e and Navara, 2001; De Simone et al., to appear], etc. DEFINITION 51. Let us denote by L2- the following class of logics: L E La- (j if a * b and if we are given an E > 0 then there is a state s E S(L) such that s(a) 2 1- E and s(b) 2 1- E. The class Lz- is interesting in its own right but it is also relevant to the study of observables as we will see later. PROPOSITION 52. C2- is a proper subclass of L2. Proof.We assume that the reader is to a certain extent familiar with the interpretation of the Greechie diagrams of a logic (see our next Figures 1,2). Let us just recall that the "points" of a Greechie diagram are interpreted as the atoms of the logic and the straight line segments (or possibly the curve segments) group together those atoms which belong to a maximal orthogonal set (Boolean block). There is a oneto-one correspondence between the states on the logic constructed and the so called weights on the Greechie diagram. (A weight is such a (nonnegative) evaluation of the points of the diagram that the sums over Boolean blocks equal to 1.) We shall not make any distinction between the weights on a Greechie diagram and the states on the corresponding logic. The precise description of Greechie diagrams may be found in [~reechie,19711 or [~chultz,19741. Let us return to the example disproving L2 c Lz-. We start with a preliminary construction (Figure 1). Claim. The logic L given by Figure 1 has the following properties: 1. L E L2-, 2. If s is a s t a t e o n L and if s(b) = 1 , then s(a) = O . Proof. Suppose that s(b) = 1 for a state s on L. Then s(fi) = 0 for any i = 1,2,3, . . . . Therefore s(d) C s(zi) 5 s(d) C s(cj). Since s(e) = 0, the left side of the last inequality equals to one. Hence s(d) C s(q) = 1 and therefore s(a) = 0. This was to prove. Observe that we have also proved that the identity s(b) = 1 for a state s implies s(a') = 1. On the other hand, if E > 0 and r E (0,l) are given, we may find a state s such that s(b) = 1- E and s(a) = r. It is easily seen that if we take two noncompatible elements p, q E L such that (p, q) # (a, b), there exists a state s on L with s(p) = 1= s(q). Hence L E Lz-.

+

+

+

Pavel Pt6k, Sylvia Pulmannov6

e

Figure 1. The desired example establishing La[Rogalewicz, 1984al.

\ L2 # 0 is now

constructed as follows

Claim. The (lattice) logic L given by Figure 2 belongs to La- but does not belong to L2. Proof. If we take noncompatible atoms (p, q) # (a, b), then one easily shows that there exists a state s on L such that sCp) = 1 = s(q). Let us consider the pair (a, b). If s(b) = 1, then s(fi) = 0 for i = 1 , 2 , 3 , . . . and therefore s(d) C s(ci) = 1. As

+

a result, s(a) 5 $ and L does not belong to Lz. On the other hand, L belongs t o La-. Indeed, if we are given a E > 0, we may simply construct a state s with s(b) = 1- E and s(fi) E ( 0 , ~ ) . An alternative proof of the previous result can be obtained by the following result, which uses the convexity of the state space. PROPOSITION 53. A logic L belongs to Lz- if and only if it satisfies the following condition: If a, b are two noncompatible elements of L and if we are given a real number T E (0,l) then there exists a state s on L such that s(a) = T = s(b). Proof. The condition is obviously sufficient. To prove necessity, let a real number

>4

< 1be given. We may suppose that T - otherwise we take up the equivalent assertion with a', b' and T' = 1- T . We shall construct two states sl,s 2 on L such

T

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Figure 2.

i)

that s l ( a ) = sl(b) = rl E (0, and s2(a) = s ~ ( b = ) 7-2 E ( r ,1). The required state s on L can then be constructed by setting

We shall first define s l . If there is a state s on L such that s(al) = 1 = s(bl), we put sl = s. If this is not the case, the assumption guarantees the existence of states s3,sq on L with s3(a1)= 1, s3(b1)= 7'3 < 1, sq(al) = 7-4 < 1, s4(bf)= 1. Let us put

Let us now consider the construction of s2. If there is a state s on L with s(a) = 1= s(b), we put s2 = s and the proof is complete. If there is no such state,

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we consider states 95, s 6 and s 7 such that sg(a) = 1, s5(b) = r5 < 1, s6(a) = rs < 1, s6(b) = 1 and, further, if we set q = max(r,r5, r6), we require s7(a) = r7 > q, s7(b) = r8 > 2. The assumptions guarantee the existence of such states. If r 7 = T,, we put s 2 = 37. If r7 < T,, then

If r,

< r7, the construction

of

92

proceeds dually. Now the proof is complete.

DEFINITION 54. An element a E L is called a support of a state s on a logic L, if for any b E L, a Ib if and only if s(b) = 0. If a is a support of s, we will write a = supp s. PROPOSITION 55. Every Jauch-Piron state on a separable lattice logic has a support. Proof. Let L be a separable lattice logic and let s be a state on L. Put so := {a E L : s(a) = 0). By Proposition 22, the element a = VsO exists in L, and a = ViENail where ai E so,i E N. Put bl = a l l bz = b l V az, . . . , b, = bn-1 V a,, . . . . Clearly, (bi)i is a nondecreasing sequence with Vi bi = a. From the JauchPiron property we obtain s(bi) = 0, i = 1,2,. . .. By a-additivity of s, s(a) = s(Vi bi) = limn,,s(bi) = 0. We will show that supp s = a'. If b E L, ~ ( b = ) 0, then b E so, and hence b Ia'. Conversely, if b Ia', then b 5 a, hence s(b) = 0.

The notion of a Jauch-Piron state can be strengthened further by introducing , RieEanovA, 1998; subadditive states (or so called valuations, see e.g. [ ~ t f i k1998; Sarymsakov et al., 19831, etc.). DEFINITION 56. Let L be a logic. A state s on L is said to be subadditive if for any a, b E L there exists c E L such that a 5 c, b 5 c and s(c) 5 s(a) s(b).

+

It is an easy exercise to show that each subadditive state is Jauch-Piron. Indeed, if s(a) = 1, s(b) = 1 then s(af) = 0, s(b') = 0. Since s is subadditive, there is c E L such that a' 5 c, b' 5 c with s(c) 5 s(af) s(br) = 0. Thus, s(d) = 1 and c' 5 a and d 5 b. If a logic has "enough" states and if each state on it is subadditive, the logic is in a sense "almost Boolean". This situation is, however, fairly subtle as the following example shows (see also [Navara and Pt&, 19881 and [Ptiik, 19981).

+

EXAMPLE 57. There is a concrete non-Boolean logic such that each state on it is subadditive. Let us indicate the construction of such an example. Let P be a set of the first uncountable cardinality. Then each state on exp P lives on a countable set a well-known result of measure theory (by exp P we mean the Boolean a-algebra of all subsets of P). In other words, if s is a state on exp P, then there is a countable subset, M, of P such that s(M) = 1. Let L be the logic of all subsets

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of {1,2,3,4) whose cardinality is even. Take {1,2,3,4) x P and let us make the latter set an underlying set of the logic in question. Let us call this logic K , K = A, where A is the following family of subsets of {1,2,3,4) x P: A set T, T c {1,2,3,4) x P, belongs to A precisely when the cardinality of the set Q = P- {p E P I the set of all w E {1,2,3,4) such that ( w , p ) E T is of an even cardinality ) is at most countable. Then A is a concrete logic (of subsets of {1,2,3,4) x P ) . Indeed, {1,2,3,4) x P belongs to A. Furthermore, it is easily seen from the definition of A that if T E A, then ({1,2,3,4) x P ) \ T belongs to A as well. Finally, let T, (i E N) be a collection of pairwise disjoint elements of A. Then U Ti E A by a simple cardinality argument. We have to show that any state on A is subadditive. Let s be a state on A. Since exp P can be embedded in A, the restriction of s to e x p P is a state on exp P and as such it must live on a countable set, S. But then s is essentially a (Boolean) state on exp S and therefore it must be subadditive. This completes the construction. 5 OBSERVABLES ON A LOGIC DEFINITION 58. An observable on a logic L is a a-homomorphism from a aalgebra C of subsets of a set R to L, that is, a mapping x : C + L is an observable if (i) x(R) = 1;

(ii) x(EC)= x(E)', E E C (ECis the complement of E in R), w (iii) x ( U z l Ed)= Vi=l x(Ei) for every sequence of C.

of pairwise disjoint subsets

The set R is called the value space of the observable x. Let us notice that if (Ei)i is any sequence of elements of C, then x ( U z l Ei) = x(Ei). This is obtained by replacing (Ei)$ by the sequence of pairwise disjoint sets (Fj)j, where Fl = El, F n = E n \ (uT~; E ) , and Ui Ei = Uj Fj. This entails that the range R(x) := {x(E) : E E C) of an observable x is a Boolean sublogic of L. A family (x,), of observables on L is called compatible if the ranges of all x, belong to the same Boolean sublogic of L. If 52 = IR and C is a suba-algebra of the Bore1 a-algebra B(B) of subsets of W,then an observable x : C -t L is called a real observable: In this case, if L is a concrete logic, then an observable is nothing but a classical random variable [~aradarajan,19851, and this is the case in many other situations [ ~ t i k19811. , If f : R + IR is a measurable function and x : C + L is an observable, then f (x) : B(W) + L, f (x)(E) = x(fP1(E)) is also an observable, which is called the function f of the observable x.

Vzl

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The theory of real observables on a logic was developed by Varadarajan [Varadarajan, 19851. The corner stone is the celebrated Loomis-Sikorski theorem. The statements in [Varadarajan, 19851 are formulated for a lattice logic but if we replace pairwise compatibility by the stronger notion of compatibility, the statements hold for arbitrary logics as well. THEOREM 59. (Loomis-Sikorski), [Varadarajan, 1985, Theorem 1.31 Let B be a Boolean a-algebra. Then there exists a measurable space ( R ,C ) , where C i s a a-algebra of subsets of a nonempty set R , and a a-homomorphism h from C onto B. THEOREM 60. [Varadarajan, 1985, Theorem 1.6, Lemma 3.161 Let x be a real observable o n a logic L. Then the range R ( x ) = { x ( E ) : E E B ( R ) ) of x i s a countably generated Boolean sublogic of L. Conversely, if B i s a countably generated Boolean sublogic of L , then there exists a real observable x such that R ( x ) = B. THEOREM 61. [Varadarajan, 1985, Theorem 1.41 Let ( R , C ) be a measurable space, and let h be a a-homomorphism from a onto a Boolean a-algebra B. Let u be a real observable with range in B. T h e n there exists a C-measurable function f : R + B such that u ( E )= h ( f -'(E)), E E B ( B ) . The function f is unique in the sense that if g i s any other function with the same properties, then { w E R : f ( w ) # g ( w ) ) belongs to the null space of h. THEOREM 62. [Varadarajan, 1985, Theorem 3.91 Let { x , : a E A ) be a n arbitrary family of compatible real observables on a logic L. T h e n there exists a set X , a a-algebra t3 of subsets of X , real-valued B-measurable functions g, on X ( a E A), and a a-homomorphism r of t3 into L such that

for all a E A and E E B ( B ) . Suppose further that either L is separable, or that A i s countable. T h e n there exists a n observable x and real valued Borel functions f, of a real variable such that V a E A, x , = fa o x . Notice that for at most two observables the condition of compatibility in Theorem 62 may be replaced by pairwise compatibility [Ramsay, 19661. Theorem 62 enables us to construct a calculus of functions of several compatible observables. THEOREM 63. [Varadarajan,1985, Theorem 3.171 Let L be a logic and x l , x 2 , . . . ,x, compatible observables on L. T h e n there exists one and only one a-homomorphism T of t3(Rn) into L such that for all E E B ( R ) and all i = 1,2,. . . ,n,

xi ( E ) = T (TF' ( E ) ) , where ~i i s the projection ( t l ,t 2 , .. .,t,) Borel function on Bn,

ti of Bn to B . If g i s any real-valued

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is an observable. If g l , gz, . .. ,gk are real valued Borel functions on Bn and yi = gio(xl,x2, . . . ,xn), then yl, y2, . . . ,yk are compatible, and for any real valued Borel function h on W k ,

where h(g1,gz, . . . ,g k ) is the function

The a-homomorphism T of Theorem 63 is called the joint observable of X I , 22,. . .,x,. In analogy to self-adjoint operators, we can also introduce the notions of an eigenvalue and the spectrum of a real observable. DEFINITION 64. A real number t is an eigenvalue of a real observable x if x ( { t ) )f 0. DEFINITION 65. Let x be a real observable on a logic L. Put

a ( x ) := n

{ : C~is a closed subset of W with x ( C ) = 1).

Then the closed set a ( x ) c W is called the spectrum of the observable x. PROPOSITION 66. The spectrum a ( x ) of a real observable x is the smallest closed subset of R such that x(C) = 1.

Proof. Since the topology of the real line W satisfies the second countability axiom, there exists a sequence of closed subsets C1,C2,.. . such that z(C,) = 1for emh n , and a ( x ) = r),"==, C,. Since x(W\C,) = 0 for all n , we have x(W \a(x)) = X(U,"=~(W \ C,) = V,"==l x(W \ C,) = 0, which entails that x ( a ( x ) )= 1. rn A number T E a ( x ) is a spectral value of x. An eigenvalue is a spectral value, the converse need not hold. LEMMA 67. A real number is a spectral value of an observable x if and only if x(U) # 0 for every open subset U c W which contains r .

Proof. Let U be an open set containing r and let x(U) = 0. Then W \ U is closed and x ( B \ U ) = 1. This yields W \ U > o ( x ) ,so that r $ a ( x ) . Conversely, assume that r $ a ( x ) . Since W is a metric space, there exist open subsets Ul,U2such that r E Ul, a ( x ) c U2 and Ul n U2= 0. Then x(U2)= 1 and rn this implies that x(Ul) = 0. DEFINITION 68. A real observable x is said to be (i) discrete if a ( x ) = { T I , r2, . . .); (ii) constant if a ( x ) = { r ) ;

Pave1 Ptdk, Sylvia Pulmannovd

168

(iii) simple if a(x) = {rl,r 2 ,

...,r,);

(iv) a proposition or 0 - 1observable if o(x)

c (0, 1).

For a E L, denote by qa the unique 0 - 1 observable for which qa({l)) = a. The observable corresponding to 0 E L, go, is the null observable. The observable corresponding to 1 E L, ql, is the unit observable. The null and unit observables are constants, and r.ql (defined by the function calculus) is a constant observable with the spectrum a(%)= {r). LEMMA 69. (i) For every a E L, q,, = f (9,) with f (t) = 1 - t (t E R). (ii) If x is an observable and XE is a characteristic function of a set E E B(R), then XE(X)= qz(E). PROPOSITION 70. The following conditions are equivalent: (i) x is a proposition observable; (ii) there exists an observable y and E E B(R) such that x = XE(Y); (iii) x2 = x. Proof. (i) & (ii): By Lemma 69, xil1(x) = x. (ii) & (iii): If x = x ~ ( y )then , (i): Put f(t) = t, g(t) = t2, t E W. We have x2 = Xs(y) = xE(y) = x. (iii) {t : f (t) # g(t)) c UrEqf -l(-oo, r)Ag-l(-oo, r), where Q is the set of rational numbers, and E A F = (Er7FC)u( E c nF) is the symmetric difference of sets. Then ~ ( f - ~ ( - o o , r ) A ~ - ~ ( - or)) o , = 0, which entails that x({t : t2 = t)) = x({O, 1)) = 1, hence x is a proposition observable.

6 PARTIAL COMPATIBILITY AND JOINT DISTRIBUTIONS OF OBSERVABLES In the sequel we will introduce the notion of partial compatibility and that of a commutator. We will then study properties of these concepts and relations between them. These results will be used in our study of joint distributions of observables. We mostly restrict ourselves to lattice logics. The notion of a commutator of two elements in an orthomodular lattice L was , The main result was that the orthomodular ideal introduced in [ ~ a r s d e n19701. J generated by the commutators of all pairs of elements of L is the smallest orthomodular ideal such that the quotient L / J is a Boolean algebra. The commutator , of a finite subset of an orthomodular lattice L has been used in [ ~ e r a n 1979; Beran, 1985; Bruns and Greechie, 1990; Bruns and Kalmbach, 1973; Greechie and Herman, 1985; Chevalier, 1984; Chevalier, 1989; Chevalier and Pulmannovi, 2000; e, Kalmbach, 19831, and [ ~ o ~ u n t k19831. The generalization of the concept of a commutator to arbitrary subsets of an orthomodular lattice L was studied in a series of papers [Dvure~enskijand Pulmannovi, 1982; DvureEenskij and Pulmannovb, 1984; LutterovB and Pulmannovi,

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1985; Pulmannovii, 1980; Pulmannovii, 1981b; Pulmannovi, 1985; DvureEenskij 1993; Ptik and Pulmannovi, and Pulmannovii, 19821, see also [~vure~enskij, 19911, the notion of partial compatibility was introduced in [~ulmannovii,1981bl. The notion of a joint probability distribution of Gudder type was introduced in [Gudder, 1968; Gudder, 19791. The subsequent developments can be followed 19821-[~vure~enskij, 1987131, [Lahti and Ylinen, 1987; Lutterovi in [~vure~enskij, and Pulmannovi, 1985;Pulmannovii, 1978;Pulmannovii, 1980; Pulmannovi, l98la; Pulmannovi, 1985; DvureEenskij and Pulmannovi, 1982; DvureEenskij and Pulmannovi, 1984; Pulmannovii and DvureEenskij, 19851, and [Pulmannovii and Stehlikovi, 19861. Heisenberg's uncertainty relations were expressed in the language of quantum logics in [Lahti, 1980; Bugajski and Lahti, 1980; Lahti and Mqczyliski, 19871. Results on the relation between uncertainty relations and total noncompatibility can be found in [~ulmannoviiand DvureEenskij, 19851 The notion of complementarity was introduced in [Lahti, 19801. There is a huge list of publications devoted to these problems, which are relevant to quantum theory see, e.g., [Gudder, 1979; Jammer, 1974; Jauch, 1968; Holevo, 1980; Mackey, 1963; von Neumann, 1932; Primas, 19811. The existence of sums of observables has already been emphasized by von Neumann [von Neumann, 19321. In some axiomatic approaches to quantum mechanics the existence of sums of bounded observables is postulated (e.g., in [Segal, 19471, and the C*-algebraic approach [Alfsen and Schulz, 1979; Emch, 1972; Haag and Kastler, 19641). The logics with the property that every two bounded observables are summable were investigated in [Gudder, 1966; Riittimann, 19851, and [DvureEenskij and Pulmannovii, 19801. Joint distributions of Urbanik type were introduced in [Urbanik, 19611 and in [Varadarajan, 19851 for observables on a Hilbert space logic. In [Gudder, 19661 a generalization of this notion to observables on a larger class of logics was studt, Urbanik, 1985; ied. For the further development see, e.g., [ ~ u ~ m g a a r1983; DvureEenskij and Pulmannovii, 1989; Pulmannovii and DvureEenskij, 19801.

6.1

Partial compatibility and the commutator

Let L be a logic, D = {-1,1), and for a E L, let a w l = a', a1 = a. DEFINITION 71. We say that a subset A of a logic L is partially compatible with respect to an element b E L (in short, A is p.c. b) if the following conditions are satisfied: (i) b tt A, i.e., b * a for all a E A; (ii) the set b A A := {b A a : a E A) is a compatible subset of L. Recall that for a subset A C L, C(A) = {b E L : b * A ) is a sublogic of L, where b * A means that b * a Va E A. It what follows, let us agree to write CC(M) instead of more correct C(C(M)).

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THEOREM 72. If for a E L a finite subset M of L is p.c. a, then CC(M) is p.c. a.

Proof. Let M = {al, a2,. . . ,a,) and let {el, e2,. . . ,ek) be a minimal orthogonal cover of M A a in [0,a]. Since a 6 C(M), a cr b for all b E CC(M). From ei * a j A a and ei a for j 5 n, i 5 k, we easily derive that ei E C(M). Hence b tt ei, i k. Therefore {{b A ei, b' A ei)f=l, b A a A (Visk ei)') is an orthogonal ' p.c.a. cover of the set {al A a, a2 A a, . . . , a, A a, b A a), which shows that M U {b) is By induction, we see that every finite subset {bl, b2,. . . , b,) c CC(M) is p.c.a, hence CC(M) B ' p.c. a.


0)(3s E V(xl,. . . , x,)) : vars(xl)vars(xz). . . var,(x,)

< E.

1. (36 > O)(VS

In the case (1) we say that the observables XI,.. . ,x, satisfy the uncertainty relation. In the case (2) we say that the observables X I , . . . ,x, do not satisfy the uncertainty relation. PROPOSITION 95. If the observables XI, . . . ,x, are compatible, they do not satisfy the uncertainty relation. Proof. Let w : B(M) + L be the joint observable for X I , .. . ,x,. By Lemma 67 (generalized in an obvious way to observables on B(Rn)), a point (TI,rz, . . . ,r,) belongs to the spectrum w(w) of the observable w iff for any 7 > 0 we have

Since S is unital, there is a state s E S such that

The properties of the joint observable imply that s ( ~ i ( ( ri 7, Ti

+ 7))) = 1 (i 4 n).

This yields, for all i (i 4 n),

The last inequality implies that the observables XI,.. . ,x, do not satisfy the unrn certainty relation. THEOREM 96. If the observables XI,.. . ,x, satisfy the uncertainty relation, then com(xl, . . . ,xn) = 0. Proof. Put c = com(xl, . . . ,x,) and suppose that c # 0. Define Sc := {s E S : s(c) = 1). Since S is unital, S, # 8. If we restrict S, to the logic [0,c], we easily verify that Sc is a unital set of states for [O, c]. The observables xi A c (i 5 n) are compatible and have the same probability distributions in every state s E Sc as the observables xi (i 4 n). By Proposition 95, the observables xl A c, . . . ,x, A c on [0,c] do not satisfy the uncertainty relation. Since the variances of the observables xi A c (i 4 n) are equal to the variances of the observables xi (i 4 n) for all states in S,, the latter observables do not satisfy the uncertainty relation. This concludes rn the proof.

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The following definition is an attempt to formulate the notion of complementarity in the frame of quantum logics. It is motivated by the notion of complementarity of observables on the Hilbert space logic L(H). On L(H) the observables P and Q corresponding to momentum and position, respectively, are complementary. DEFINITION 97. We say that two observables x and y are complementary if for any bounded sets El F in B(W) such that w(x) (f El w(y) (f F we have x(E) A y(F) = 0. In the following propositions we list elementary facts concerning complementarity. The simple proofs are left to the reader. PROPOSITION 98. Two compatible observables are complementary if and only if at least one of them is a constant. PROPOSITION 99. Let a, b (a, b # 0 , l ) be two elements of L and let q,, qb be the cowesponding proposition observables such that q,({l)) = a, qb({l)) = b. Then q, and qb are complementary if and only if they are totally noncompatible. PROPOSITION 100. If the observables x and y are complementary and noncompatible, then they are totally noncompatible. The following example shows that the converse of Proposition 100 does not hold. Consider the Hilbert space W3. Notice that there exists no pair of nontrivial complementary observables. Let B1 = {el, e ~es) , and B2 = { f 1, f2, f3) be two bases of W3 with B1 n B2 = 0. Let [el denote the one-dimensional subspace generated by e E W3. Define two observables x and y on L(W3) as follows. Choose three real numbers TI, r z , T-3and put x({ri)) = [ei], y({ri)) = [fi] (i = 1,2,3). Then x and y are totally noncompatible but not complementary.

6.4

Sum logics and joint distributions of Urbanik type

The notion of a sum logic has its motivation in the quantum logic L(H) of closed subspaces of a (complex, separable) Hilbert space H. By the spectral theorem, there is a one-to-one correspondence between the real observables on L(H) and the self-adjoint operators on H . In particular, for every two bounded observables, their sum is defined as the observable corresponding to the operator sum of the corre sponding bounded self-adjoint operators. In certain cases the sums of unbounded operators exist, too. We will try to find a generalization of this phenomenon in the study of "sum logics". We note that in the definition of a sum logic L we need not require L to be a lattice logic. Let us recall the following notions (compare with Definition 44). A set S of states on L is said to be rich if the inclusion {s E S : s(a) = 1) C {s E S : s(b) = 1) implies a 5 b. A set of states S on L is said to be order determining if s(a) 5 s(b) for all s E S implies a 5 b. For a set S of states, we denote by conv(S) the a-convex envelope of S. Let L be a logic and let S be a rich a-convex set of states on L. For a real observable x on L, set V(x) := {s E S : s(x2) < 0 0 ) . In view of Schwarz

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inequality, s ( x ) ~5 s(x2), and since var, (x) = s(x2) - s ( x ) ~we , have V(x) = {s E S : var, (x) < m}. PROPOSITION 101. [Gudder, 19791 Let L be a logic and let S be a a-convex set of states on L which is rich. Let x be a real observable on L. Then x is bounded if and only if V(x) = S.

Proof. If x is bounded, then x2 is also bounded, hence V(x) = S. Conversely, let x be unbounded. We will show that there is a state s E S such that s(x2) < m does not hold. Let rn E w(x) be numbers such that 2n+2 - 1 > lrnl > 2,+l (n E N). Let U, be disjoint open balls in W with diameter less that 1 and with centers at r,. Put a, = x(U,). Since any Un is open and contains an element from the spectrum of x, we have a, # 0. Let s, E S be such that s,(a,) = 1. Since the elements ai(i E N) are orthogonal, we obtain sj(ak) = 0 for j # k. The state s = CjEN 2-jsj belongs to S. Assume that s(x2) < m . Then

which is absurd.

H

DEFINITION 102. We say that the real observables X I , (or that their sum exists) if

22,

.. . ,x, are summable

(i) the set V(xl,x2, . . . ,x,) := Uisn V(xi) is order determining; (ii) there is a unique real observable z such that V(z) > V(xl ,x2, . . . ,x,) and s(z) = ~ ( $ 1 ) s ( x ~ ) . . . s ( z ~for ) all s E V(xl, 5 2 , . . . ,5,).

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The observable z is called the sum of the observables xl, x2, . . . ,x,, and we write z = x 1 + x 2 + ...+a,. DEFINITION 103. A pair (L, S ) , where L is a logic and S is a a-convex rich set of states, is called a sum logic if the following two conditions are satisfied: (i) every two bounded observables are summable; (ii) every two compatible observables are summable. The above definition implies that summability and the sum of observables xl,x2,. . . ,x, are invariant with respect to permutations of the set {1,2,. . . ,n} and with respect t o the mappings xi + rxi (i 5 n) for every r E W. Moreover, if X I , x2,. . . ,x, are summable and if the sums yl = XI x2 . . . xk: and y2 = xk+l x, exist, then XI + x 2 + - - - + x n = yl +y2. Notice also that if xi,i 5 n are compatible and xi = u. fsrl(i 5 n), then 2 1 . xn = u.(fl . . . fn)-l. We will see that the logic L(H) with S = S(L(H)) is a sum logic, and so are the logics of projections in a von Neumann algebra. To our knowledge, a characterization of sum logics is not known.

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Proposition 99 and Proposition 101 imply that every two bounded observables have a bounded sum. This allows us to prove the following statements. PROPOSITION 104. Let (L,S) be a sum logic. Then (i) if x, y are bounded observables, then x = y if and only if s(x) = s(y) (that is, L has the Uniqueness property); (ii) L is a lattice; (iii) every state s E S has the Jauch-Piron property (that is, for every s E S , if a, b E L and s(a) = 1 = s(b), then s(a A b) = 1). Proof. (i) Suppose that s(x) = s(y) for all s E S. Let qo be the proposition observable qo({O)) = 1. Then the observables x qo and y qo are sums of the observables x and qo. Since the sums are unique, we have x qo = y qo. Hence x = y. (ii) Let a, b be elements of L and let q,, qb be the corresponding proposition observables. We will show that (q, qb)({2)) = a A b. From s((q, qb)({2))) = 1 it immediately follows that s(q, qb) = 1, hence s(a) = 1 = s(b). Since S is rich, it follows that (q, qb)({2)) is a lower bound of a, b. Let c E L be such that c 5 a, b. Then the equality s(c) = 1implies s(a) = 1 and s(b) = 1. Hence, s(q,+qb) = s(a) +s(b) = 2. Since 2 is the maximal point of the spectrum w(qa+qb), we see that s(q, qb)({2)) = 1. This implies that c 5 (q, qb)({2)). We have shown that the infimum a A b exists in L for every a, b E L, i.e., L is a lattice. (iii) Suppose that s(a) = 1 and s(b) = 1 for s E S. By the proof of (ii), we conclude that s(a A b) = 1. Thus s has the Jauch-Piron property.

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The notion of another type of joint distribution is based on the CramCr-Wold theorem in classical probability theory. Let p be a probability measure on B(Wk). For a E Wk, define ha : Wk + W by putting h,(r) = a.r = CiCkairi. If a is a real number, then the set {r : a.r 5 a) = P,(a) is a closed half-space in lRk. Further, we have p(P,(a)) = p.h;l(-oo,cr]. We recall that the CramCr-Wold theorem states that if pl and pz are probability measures on B(Wk) such that pl (Pa(a)) = pz(Pa (a) for every a E Wk and every a E W, then pl = p2. DEFINITION 105. We say that the observables X I , xz, . . . ,xn on a sum logic have a joint distribution of Urbanik type in a state s if the following two conditions are satisfied: aixi exists for any a (i) the sum Xi Un M,). Thus, L(H) is a a-complete lattice. The mapping M -+ M' is an orthocomplementation. Indeed, the inclusion M C N obviously implies N' c MI. Further, M A M' = 0, because x E M and x E M' yields (x, x) = 0, which means x = 0. The inclusion M C M u is obvious. Assume that x E M u \ M. By Theorem 109, x = y z with y E M and z E M'. We have z = x - y E M u , and therefore z E M u A MI. Thus z = 0, and hence M = Mu. Finally, assume that M c K ~ But and M ' A K =O. I f x K~ , thenwe h a v e x = y + z for y~ M a n d z MI. z = x - y E K , and, therefore, z = 0. Thus, M = K , and we have verified the orthomodular law.

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By an operator on H we mean a linear mapping A : H -+ H. The n o r m of an operator A is defined by IlAll = sup{IIAxII : x E H, IIxII = 1). An operator A is bounded if the norm of A is finite. If A is bounded, the adjoint operator A* of A is defined by the equality (Ax, y) = (x, A*y) for any x, y E H . A bounded operator A is self adjoint if A = A*. A projection operator (also called a projection or a projector) is a bounded operator P such that P = P2= P*. THEOREM 111. To every closed subspace of H we can associate a unique projection, and every projection is associated with a unique closed subspace of H .

Proof. Assume that N E L(H). The projection associated with N , denoted by pN,is defined as follows: p N x = y, where y is the unique vector of N given by the decomposition x = y + z in the sense of Theorem 109. On the other hand, if P is a projection, the set Np = {x E H : P x = x) is the subspace of H associated with P. One can simply verify that pNis a projection, Np is a closed subspace, rn and that Nplv = N a n d pNp= P. The set of all projections on H can be endowed with a partial ordering by writing P 5 Q if P Q = Q P = P. The following statements are well known and easy to prove (e.g., [Halmos, 19511).

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PROPOSITION 112.

pN be the corresponding (i) Let M, N be closed subspaces of H and let pM, = pNpM, i.e., projections. Then M c N if and only if pM= pMpN pM5 pN.

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(ii) M IN if and only if pMpN = pNpM = 0 if and only if pM pN is a projection. (iii) M, N are compatible in L(H) if and only if pMpN = PNPM (iv) For two projections P, Q, the operator Q - P is a projection if and only if p I Q(v) P Q is a projection if and only if P Q = Q P . (vi) If P is a projection, then 1- P is also a projection, and N 1 - ~= Nb. The states on the logic L(H) are characterized by the famous Gleason theorem [Gleason, 19571 stated below. THEOREM 113. (Gleason theorem) Let H be a separable Hilbert space with dim H 2 3. If s E S(L(H)) is a pure state on L(H), then there exits a unit vector v, E H such that s ( P ) = (v,, Pv,) (P E L(H)). Moreover, every state on L(H) can be written as a a-convex combination of pure states such that the unit vectors corresponding to them are mutually orthogonal. Gleason's theorem implies that the pure states s on L(H) are in a one-to-one correspondence with unit vectors v, in H (i.e., v, E H, Ilv,l\ = 1) given by s(P) = ( P v , , ~ , ) . Therefore pure states are also called vector states. Moreover all states on L(H) are in a one-bone correspondence with the positive self-adjoint traceclass operators with unit trace (so-called density operators). That is, to every state s there corresponds a positive operator T,= CiEN c ~ P [ ~ ~ I ,where CiEN cj = 1, and {vj : i E N} is a complete orthonormal set of vectors in H. We then have that s(P) = tr(T,P) = CiEN ~ ( P v jvj). , Let us note that Gleason's theorem also holds for separable real and quaternionic Hilbert spaces with dimension at least three and it can be extended to certain nonseparable Hilbert spaces (see [~vure~enskij, 1993; Hamhalter, 20031). Let ( 0 , C) be a measurable space. A spectral measure is a mapping E : C + L(H) such that the following conditions are satisfied: (i) E ( 0 )

=

1;

(ii) if S1,S2E C and S1n S2= 0, then E(Sl)

IE(S2);

(iii) if Sj, i E M is a collection of pairwise disjoint sets of C, then E(UjENSi) = CiENE(Si), where the sum on the right converges in the strong operator topology.

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In other words, a spectral measure is an observable E : C 4 L(H). If C = B(W), then E is a real observable. The real observables on L(H) are characterized by the spectral theorem. Before stating it, let us first generalize the notion of bounded operator. Let D be a dense subspace of H and let T : D + H be a linear mapping. Call T again an operator on H. Then one can define the operator T* adjoint to T such that the following conditions are satisfied: The domain of T* is determined by the requirement D(T*) = {y E H : there is a unique y* E H such that (Tx, y) = (x, y*) for any x E D(T)), and we define T*y = y* (y E D(T*)). An operator T is called self-adjoint if T = T*.

THEOREM 114 (Spectral theorem). For any self-adjoint operator A on a Hilbert space H there exists a unique observable pA: B(B) + L(H) such that (i) the domain D(A) of A consists of all vectors x E H for which the integral J X2 ( p A(dX)x, x) converges, (ii) for any x E D(A) and any y E H we have (Ax, y) = 5 x(P*(~x)x,y). Conversely, for any real observable E there exists a unique self- adjoint operator

A such that E = pA. For the proof of the spectral theorem see, e.g., [Halmos, 19511. The spectral measure corresponding to a self-adjoint operator A is also sometimes called the spectral resolution of A. The spectrum of an operator A is defined as the set of all complex numbers c such that the operator A- cl has no bounded inverse mapping. It can be proved that the spectrum of a self-adjoint operator A coincides with the spectrum of the corresponding observable pA. Let s be a state on L(H) and let T be the corresponding density operator. Let A be a self-adjoint operator and let pAbe the corresponding observable. Then for the expectation of A (equivalently, of p A ) in the state s we have

if the integral converges. Let us conclude this section by showing that the Hilbert space logic is a sum logic in the sense of Definitions 102, 103. This fact is important in quantum axiomatics. By the spectral theorem, if A is a self-adjoint operator, we can identify it with the corresponding observable, and hence view A as a mapping from B(W) to L(H). If f : W + W is a measurable function, then f (A) = A.f-I is an observable too (for example,A2 = A.f -', where f (t) = t2). Denote by D(A) the domain of a self-adjoint operator A. Recall that, in case A is unbounded, D(A) is a dense subset of H . Moreover, we can write D(A)

J t z ( ~ ( d t )u)~ ,< m )

=

{u E H :

=

{ U E H : s,(A2)

< m).

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PROPOSITION 115. Let Sv be the set of all vector states on L ( H ) . Then S C Sv is rich for L ( H ) i f and only i f S = Sv. Proof. Richness of Sv is obvious. If S # Sv, then there is a vector v E H such that sv 4 S . Let [v]be the subspace generated by the vector v . Then the equality s,([v]) = 1 holds for no s, E S , contradicting the hypothesis that S is rich.

COROLLARY 116. Let A be a self-adjoint operator. Then the following statements are equivalent: (1) the set S ( A ) of all vector states corresponding to the vectors of D ( A ) is rich for L ( H ) ; (2) D ( A ) = H ; (3) A is bounded. PROPOSITION 117. For a linear subspace K of H , write S ( K ) = {s, : u E K , llull = 1). Then K is dense i n H i f and only i f S ( K ) is order determining for L(H)Proof. Let K be dense. Then if s,(P) 5 s,(Q) for all unit vectors in K , we infer that ( P v ,v ) 5 (Qv,v ) for all v E H , and hence P 5 Q. This shows that S ( K ) is order determining. Conversely, let KO be a subset of H be such that the set S ( K 0 ) = {s, : u E K ) is an order determining set of states. Let K l be the linear subspace generated by KO. Obviously, S ( K l ) is order determining for L ( H ) . Let K be the closure of K 1 in H . Then for every u E KO we have s,(K) = 1. Since S ( K o ) is order determining, we have K = H .

Observe that if s = CiEN aisi and s(A2) < CO, then si(A2) < ca for all i E W. Moreover, if P, Q E L ( H ) and if si(P) 5 si(Q) for all i, then s ( P ) 5 s(Q). This implies that the set {s E S ( L ( H ) ) : s ( A 2 ) < ca) is order determining for L ( H ) if and only if the set of vector states {s, : s,(A2) < ca) is order determining for L(H). If A is a self-adjoint operator, its domain D ( A ) is dense in H . If A , B are selfadjoint operators, then their sum A B is defined on D ( A ) n D ( B ) . It is known that A B need not be self-adjoint. But if at least one of A, B is bounded, then A B is self-adjoint (see e.g. [ ~ u n f o r dand Schwarz, 19571, [~ru~ovecki, 19711). If A , B commute, i.e. if they correspond to compatible observables, then we have by functional calculus of compatible observables that A B is self-adjoint. Let us finally describe an algebraic construction in connection with sum logics. Let x , y be two bounded real observables on a sum logic. The so called Segal product is defined by 1 x o y = z [ ( x + ~ ) 2 (- X - Y ) ~ ] .

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In [ ~ u d s o nand Pulmannovb, 19931, the following statement was proved. THEOREM 118. Let L be a sum logic such that on the set of Ob(L)of all bounded observables on L , the Segal product is distributive (with respect to the sum of observables). Then Ob(L)admits the structure of a Jordan algebra.

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8 JOINT DISTRIBUTIONS ON THE HILBERT SPACE LOGIC The main result of this section is the statement that observables having a joint probability distribution of Gudder type in a state s also have a joint probability distribution of Urbanik type in the state s, provided that all required sums exist. Let L(H) be the logic of a separable Hilbert space H (real or complex) with dim H 2 3. Let B(M) be a Bore1 a-algebra of subsets of a separable Banach space M (later on we will restrict ourselves to the real observables). Recall briefly our notations. If C is a closed subspace of H , then pCdenotes the corresponding projection. For a vector v, the symbol [v] denotes the one-dimensional subspace generated by v. If llvll = 1, then s, : C + (v, pCv) denotes the corresponding state on L(H). By Gleason's theorem, for every state on L(H) there is a density ~ ) that T is positive, self-adjoint and operator T with s(C) = s ( p C ) = ~ T ( T P such has a unit trace. In agreement with the preceding sections, we write, for all d = (dl, dz, . . . ,d,) E Dn, D = (-1, I), and all F E L(H), Fdi = F provided di = 1, and ~~i = F' provided di = -1. THEOREM 119. Let v be a unit vector in H. The observables X I , xz, . . . ,xn on L(H) have a joint distribution of Gudder type in the state s, zf and only if for any El,Ez,.. .,En E B(M) and any permutation p of the set {1,2,. . . , n ) the following equality holds: (8) p x l ( E 1 ) . . . p ~ n ( E n=) P~X P ( ~ ) ( E P. .( .~p)~ p ( n ) ( E ~ ( n ) ~ . For the proof of this theorem we need some simple lemmas the proofs of which are left to the reader. LEMMA 120. If the equality (8) holds for a vector v E H , then ~ X ~ ( E I ) A . . . A X ~=( E pzi(E1). ~), .. p ~ n ( E n ) ~ . LEMMA 121. Let {vi : i E N) be an orthonormal set of vectors. If llv112 = CienI(", vi)I2 for a vector v E H , then v = CiEn(v,vi)vi. LEMMA 122. Let VI, Vz, for a d € Dn. Then

.. .,Vn E L(H) and let v E H, v # 0.

pVp(1)

*

. .pVp(n),

Let v E Ai z is interpreted as a proposition in L asserting that y "implies" z. A Heyting effect algebra is a lattice-ordered effect algebra L equipped with a binary operation > such that (L, ) is a Heyting algebra and C(L) = {x 3 0 I X E L).


pER

v

then Dx # 0 iff X is an eigenvalue of A. Furthermore, if X is an eigenvalue of A, then Dx is the projection onto the A-eigenspace of A. As usual, we denote by Q the ordered field of rational numbers. The subfamily is called the rational spectral resolution for A. By properties of (i) and (ii) in Example 134, for any p E B,we have

P,, =

A

PA,

PkT(p,) E U for all k no and all T E K . A combined extension of the vector-valued commutative result of Graves and Ruess and Akemann's noncommutative compactness criterion has been obtained i Hamhalter, to appear]. by E.Chetcuti and J.Hamhalter in [ ~ h e t c u tand

>

>

THEOREM 32. Let K be a subset of B,(M, X), where the latter is equipped with the topology of pointvise convergence on elements of M. Then K is relatively compact if, and only iJ K(x) is relatively compact for each x E M and K is uniformly a-additive. Relative compactness of sets of measures has another aspect - the existence of a control functional. The celebrated result of R. G. Bartle, N. Dunford and J. Schwartz [ ~ a r t l eet al., 19551 says that a set K in an L1-space is relatively compact in the weak topology exactly when K is absolutely uniformly continuous with respect to some positive measure. In order to discuss this result in the context of operators algebras we need to introduce some notions. Let X be a locally convex space and T be a completely additive linear map from a von Neumann algebra M into X . We say that T is absolutely continuous with respect to a normal positive functional I) on M (in symbols T 0 such that T(p) E U whenever p is a projection in M with I)@) A set K of linear maps mapping M into a locally convex space X is said to be uniformly absolutely continuous with respect to a normal positive functional I) on M if for each neighbourhood U of 0 in X there is a 6 > 0 such that T p E U whenever T E K and p is a projection in M with I)(p) < 6. In the case of scalar functionals absolute continuity cp 0 there is a 6 I ) ( x * x + x x * ) ~< 6.

> 0 such that

Icp(x)l

<E

whenever x E MI and

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314

A variant of the equivalence between (i) and (iii) holds also for uniform absolute continuity. One of the remarkable results of noncommutative measure theory is Akemann's characterization of weakly compact sets in the preduals of von Neumann algebras. THEOREM 34 (Akemann). Let K be a bounded set in the predual of a von Neum a n n algebra M. K is relatively compact i i and only .if, K is uniformly absolutely continuous with respect to some positive normal state o n M. If M is abelian, then M, is isomorphic to a L1-space and the previous result reduces to the result of R. G . Bartle, N. Dunford and J. Schwartz [Bartle et al., 19551. As we will see soon the characterization of weak compactness of scalar functionals in terms of uniform a-additivity has a full vector-valued generalization. In contrast, Theorem 34 cannot be directly generalized to vector measures. In order to show this fact we need some additional notation. Let K C Bca(M,X ) and U be a convex, closed, circled neighbourhood of 0 in X. Then we put

where U0 is the polar of U (i.e. U0 = {f E X* I If (x)l 5 1 for all x E U)). If X is a normed space, we will simply write I? to denote KX,.The following two results may be found in [Chetcuti and Hamhalter, to appear]. THEOREM 35. The following conditions are equivalent for K

c Bca(M,X):

(i) K is uniformly a-additive and bounded. (ii)

kU i s

(iii)

Krr i s uniformly

bounded and uniformly a-additive for each closed, circled, convex neighbourhood U of 0. absolutely continuous with respect to some each closed, convex, circled neighbourhood U of 0.

E M$ for

Moreover, if X is metrizable, then we can add the following condition: (iv) K i s uniformly absolutely continuous with respect to some $ E M$. COROLLARY 36. If K c B,(M,X) is relatively compact in the topology of pointurise convergence o n elements of M, where X i s metrizable, then there i s $ E M, such that K is uniformly absolutely continuous with respect to $. The reverse implication in the foregoing theorem is not valid. Indeed, let M be a von Neumann algebra and H a Hilbert space of infinite dimension with the unit sphere Sl(H). Fix a normal state $ on M and define a collection of H-valued in the following way completely additive maps, K = (Tc)EES1(H),

It is clear that K is uniformly absolutely continuous with respect to $. However, if $(x) # 0, then K(x) = $(x) Sl(H) is not a compact subset of H. In view

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of Theorem 31, this means that K is not relatively compact in the pointwise convergence topology. Akemann's characterization of weak compactness in terms of uniform a-additivity has the following interesting aspect. Since uniform aadditivity depends on decreasing (and so mutually commuting) projections, we see that the weak compactness is commutatively determined in the sense that a set K in the predual of a von Neumann algebra M is weakly relatively compact if, and only if, the restriction of K to any abelian von Neumann subalgebra of M is weakly relatively compact. For a long time after Akemann's result it had not been clear whether the same holds for duals of C*-algebras. This problem was settled affirmatively in a deep result due to H. Pfitzner [1994]. (Let us recall that positive elements x, y in a C*-algebra are called orthogonal if x y = 0.) THEOREM 37 (Pfitzner). Let A be a C*-algebra and K a bounded subset of the dual A* of A. Then K is weakly relatively compact iJ and only iJ limn,, supfEh-If (a,)l = 0 whenever (a,) is a sequence of mutually orthogonal self-adjoint elements in A. The previous result is a noncommutative version of Grothendieck's criterion of compactness for C(S1) spaces: A bounded set K in the dual of a commutative C*-algebra C(Q) is not weakly relatively compact if, and only if, there is a sequence of disjoint open sets (0,) such that K which does not converge uniformly to zero on (0,). In connection with Grothendieck's compactness criterion A. Pelczynski [Pelczynski, 19621 introduced and studied the so-called property (V). This definition, originally motivated by studying weakly compact operators, singles out those Banach spaces to which Grothendieck's compactness criterion can be applied. A series Enx, in a Banach space X is called weakly unconditionally Cauchy if EnIf (x,) 1 < oo for all functionah f E X*. A Banach space X has the property (V) if for any (bounded) set K c X* which is not compact in the weak topology there is a weakly unconditionally Cauchy sequence (x,) c X such that supf E~ If (x,)l does not converge to zero as n + oo. Equivalently, X has property (V) if, and only if, every weakly unconditionally converging operator defined on X is weakly compact. (An operator T : X --+ Y between Banach spaces is called weakly unconditionally converging if it sends weakly unconditionally Cauchy sequences in X to unconditionally converging series in Y. T is said to be weakly compact if T(XI) is a weakly compact set in Y.) As a consequence of Pfitzner's theorem we see that all C*-algebras have property (V). A Banach space X is said to have the Grothendieck property if each sequence (f,) in its dual X * which converges in the weak* topology converges also in the weak topology on X*. Since any dual space with property (V) enjoys the Grothendieck prop erty, an important corollary of Pfitzner's theorem says that any von Neumann algebra has the Grothendieck property. This is a striking result on the sequential topological structure of the duals of von Neumann algebras. (Before Pfitzner's theorem it had been known that weak and weak* convergence coincide for sequences of positive functionah [Akemann, 19671.) The fact that compactness is commutatively determined in the duals of operator algebras has an interesting

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bearing on automatic compactness of certain Banach space-valued linear maps on operator algebras. It is a classic result of A.Pelczynski that a Banach space X has property (V) if, and only if, for any bounded operator T from X to a Banach space Y the following dichotomy holds: either T is weakly compact or there is a subspace Z of X isomorphic to Q such that T restricts to an isomorphism on Z. In particular, any bounded map from a C*-algebra to a Banach space not containing any copy of Q is automatically weakly compact. (Theorem IV.2 in [Akemann et al., 19721). It provides a more general explanation of the classic result of Grothendieck [Grothendieck, 19531 to the effect that any bounded linear map from the space C(K) into a weakly sequentially complete Banach space is automatically weakly compact. Another important aspect of weakly compact operators studied till today in the noncommutative context is the Dunford-Pettis property. A Banach space X is said to have the Dunford-Pettis property if for each Banach space Y each weakly compact operator from X to Y is completely continuous, that is, it carries weakly convergent sequences to norm convergent sequences. It was shown by Dunford and Pettis that L1-spaces have this property. A remarkable result of Grothendieck says that C(K)-spaces, alias commutative C*-algebras, enjoy this property too. Subsequent investigations of this phenomenon for general C*-algebras [Chu and Iochum, 1990; Hamana, 19771 discovered that a C*-algebra A has the Dunford-Pettis p r o p erty if, and only if, every irreducible representation of A is finite-dimensional. Moreover, it was proved that preduals of von Neumann algebras, alias noncommutative L1-spaces, have the Dunford-Pettis property if, and only if, the von Neumann algebras in question are finite Type I [ ~ u n c e1992; , Chu and Iochum, 1990; Hamana, 19771. The Dunford-Pettis property has the following equivalent form: A Banach space X has the Dunford-Pettis property if, and only if, whenever (x,) and (en) are weak null sequences in X and X*, respectively, then e,(x,) --+ 0. In [F'reedman, 19971 the secalled alternative Dunford-Pettis property was introduced: A Banach space X has the alternative Dunford-Pettis property if whenever a sequence x, + x weakly in X with IIxnll = 11x11 = 1, for all n, and (en) is a weak null sequence in X*, then e,(x,) + 0. This property is weaker than the Dunford-Pettis property. It was proved by L. J. Bunce and A. M. Peralta that the Dunford-Pettis property and the alternative Dunford-Pettis property are equivalent for all C*-algebras and that a von Neumann algebra is of Type I if, and only if, its predual has the alternative Dunford-Pettis property [Bunce and Peralta, 20021.

3.3 Noncommutative convergence theorems In this section we focus on various convergence theorems arising in noncommutative measure theory. A prominent convergence theorem in classical measure theory is the Vitali-Hahn-Saks Theorem. We will deal mainly with its completely additive version [ ~ u n f o r dand Schwartz, 19881. THEOREM 38 (Vitali-Hahn-Saks).

Let A be a complete algebra of subsets of

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a set !2 and ca(A) the set of all (complex) completely additive measures o n A endowed with the topology of pointwise convergence on elements of A. Suppose that K C ca(A) is a relatively compact set such that every element of K i s absolutely continuous with respect to a positive completely additive measure p o n A. Then K i s uniformly absolutely continuous with respect to p.

Any functional p on a C*-algebra A can be decomposed canonically as p = ip3 - ip4, where p l , pz, ( ~ 3p , 4 are uniquely determined positive functional~such that p l and pz are orthogonal and p3 and cp4 are also orthogonal. We shall denote by [p]the sum pl pz p3 p3. Given a positive functional $, p is said to be strongly absolutely continuous with respect to $ if [p] is absolutely continuous with respect to $. The first noncommutative generalization of the Vitali-Hahn-Saks theorem was given by A. Aarnes [1966] and C. A. Akemann [I9671 t o the effect that a relatively weakly compact set K in the predual of a von Neumann algebra M is uniformly absolutely continuous with respect to a positive normal functional $ on M whenever each p E K is strongly absolutely continuous with respect to $. Since strong absolute continuity coincides with absolute continuity for abelian algebras, Aarnes's and Akemann's advances were a direct generalization of the classical Vitali-Hahn-Saks Theorem for positive measures. However, these two concepts are not equivalent in the noncommutative case no rooks et al., 20041. It was therefore not clear whether a direct generalization of Vitali-Hahn-Saks Theorem to absolute continuity of nonpositive measures could also be obtained. The answer has was given in the negative by E. Chetcuti and J. Hamhalter who provided the following counterexample [Chetcuti and Hamhalter, to appear]. p1 - p z

+

+ + +

EXAMPLE 39. Let (Jn)r=l be an orthonormal basis of a separable Hilbert space H . For any two unit vectors J and q in H , we write we,, to denote the linear functional on B(H) defined by wC,,(x) = (xJ, q) (x E B(H)). Let $ = C r = z +wendn. Consider the sequence ( W , C , , ~ , + ~We ) ~ prove = ~ . that this sequence is weakly convergent and pointwise absolutely continuous with respect to $ but that it fails to be uniformly absolutely continuous with respect to $. If $(e) = 0 for some projection e of B(H), then e must be the projection of H onto the span{Jl). Thus, w ~ ~ , ~ ~=+0 for ~ ( all e )lc and so w < , , ~ 0 define the transformation, h,, of h as follows: h, = h ( l E h)-l . Now by putting

+

wh (x) = sup w ( h i / 2x h;j2)

,

€>O

we obtain another a:-invariant [1973] the converse:

weight on M. Pedersen and Takesaki proved in

THEOREM 46 (Pedersen-Takesaki Radon-Nikodym). Let M be a von Neumann algebra and w a faithful semijinite normal weight o n M with the modular group (u?)tEw. For each semijinite normal weight cp on M invariant with respect to there i s a unique (unbounded) positive operator h afiliated with M Wsuch that cp=Wh.

If the control measure w is a trace, then the modular group of w is trivial and so every normal semitinite weight cp satisfies conditions of the previous theorem. In particular, if a von Neumann algebra M admits a faithful normal semifinite trace r , then any normal positive functional cp on M is of the form cp(x) = r h . This e , Segal, 19531. result is due to H. Dye and I. Segal [ ~ ~1952; J. Tisher showed in [1979] that the Radon-Nikodym derivative should be expressed in terms of affiliated spectral measures rather than affiliated operators. It

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allows one to formulate Pedersen-Takesaki Radon-Nikodym theorem for not neces sarily semifinite weights. We shall briefly discuss this approach. Let w be a faithful semifinite normal weight on M. Let E : B([O,oo]) + P ( M W )be a a-additive spectral measure on the algebra of Borel subsets of [0,oo]. For each x E M+, E induces a (positive) Borel measure, p,,,, on B([O, oo]) by

for each Borel set X putting

c [0,oo]. Another weight, WE,

can now be defined on M+ by

Tisher's result says that any normal weight cp on M invariant with respect to the modular group of w is of the form WE described above. Let us recall that Radon-Nikodym theorem is far from being true for general weights. The information on the mudular groups have to compensate noninvariance of general weight with respect the unitary group. Even Sakai's RadonNikodym theorem for linearly ordered functionals does not hold for weights. It was shown that if cp 5 w where cp is a normal semifinite weight and w is a faithful semifinite normal weight then $ = w(h h) with unique h provided that w is finite or majorized by semifinite normal trace; but that it need not hold in general [~edersenand Takesaki, 19731. An important Radon-Nikodym type principle frequently used in operator theory is the fact that if w is a positive functional on a C*-algebra A with the G.N.S. then the map h E (rw(A)+)i4 c p h E A; given by representation rW,

is an order preserving bijection between the positive part of the unit ball of the commutant rw(A)' and the set of positive functionals linearly dominated by w. This aspect has been developed by S. Gudder in the context of general *-algebras [Gudder, 1979bl. He described the position of two positive functionals on a *algebra in terms of their (abstract) G.N.S. representations which allowed him to obtain a quite general form of the Radon-Nikodym derivative.

3.5 Noncommutative Ly apunov theorems Another distinguished principle of classical measure theory and its applications is Lyapunov's theorem. THEOREM 47 (Lyapunov). If (R,A) is a measure space and p : A + Cn is a nonatomic vector-valued measure, then the range of p is convex and compact. The first noncommutative generalization of Lyapunov's theorem was obtained by N.Azarnia and J. D. M. Wright [1982]. After that C. A. Akemann and J. Anderson [I9911 established many variants of Lyapunov's theorem for von Neumann algebras in a nice and extensive treatment [Akemann and Anderson, 19911.

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THEOREM 48 (Noncommutative Lyapunov Theorem). Let M be a von Neumann algebra which does not contain any nonzero minimal projection and let Q be a weald*-continuous linear map of M into (Cn. For each positive element a in the unit ball of M there is a projection p E M such that *(a) = Q(p). In particular, if M is nonatomic and does not contain any Type I2 direct summand, then according to Generalized Gleason's theorem the measure-theoretic form of Lyapunov's theorem follows: Any completely additive bounded measure on P ( M ) with values in a finite-dimensional space, where M is a nonatomic von Neumann algebra without Type Iz direct summand, has a compact convex range. If we drop the assumption that the range of 9 is finite-dimensional, the result is not valid even in the commutative case. However, if we assume that the map 9 is singular, then the principle holds for infinite-dimensional ranges as well. (Let us recall that a linear map 9 from a von Neumann algebra M into a normed space X is called singular i f f o 9 is a singular functional on M for each f E X*.) THEOREM 49 (Singular Lyapunov Theorem). Suppose that M is a a-finite algebra such that the centre of the finite part of M i s finite-dimensional and that X i s a normed linear space whose dual space is weak*-separable. If 9 is a singular linear map of M into X , then for each positive element a in the unit ball of M there is a projection p in M such that @(a) = Q(p). 4 NONCOMMUTATIVE PROPERTIES OF MEASURES

In the previous sections we saw that many properties of measures on classical measure spaces transfer to noncommutative case, although the proofs require essentially new ideas. In this part we shall focus on typically noncommutative results which have no analogy in standard measure theory. All the properties we are going to study have one common feature: they are automatically satisfied in the commutative case but they have surprisingly strong consequences in the noncommutative context.

1

Subadditivity and traces

Any positive (possibly unbounded) measure p on a a-algebra (a,A) is subadditive, i.e. p(A U B) 5 p(A) p(B), whenever A, B E A. Analogously, a measure p : P ( M ) -+ [0, oo] defined on the projection lattice of a von Neumann algebra M is called subadditive, if

+

In contrast to the classical case, a proper noncommutative measure is subadditive if, and only if, it enjoys the distinguished property of being invariant with respect to the unitary group. Moreover, it turns out that an improved version of Gleason's theorem holds for subadditive measures not requiring the exclusion of the central position of Type I z .

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THEOREM 50. Let p be a subadditive state on P(M), where M is a von Neumann algebra. Then p extends uniquely to a tracial state on M . Let us recall that a positive measure p on the projection lattice P ( M ) is called semifinite if there is an increasing net of projections, (e,), such that e, /" 1 and p(e,) < OCJ for all a. The following result is a Gleason type theorem for positive subadditive measures with infinite values. THEOREM 51. Let p be a completely additive semifinite subadditive measure on the projection lattice of a von Neumann algebra M. T h e n p extends uniquely to a normal semifinite trace o n M. As a consequence, the previous theorems provide a lattice-theoretic characterization of the traces. They have been established (in the more general context of JBW algebras) by L. J. Bunce and J. Hamhalter in [1995]. Let us notice that there are many characterizations of traces - see e.g. [ ~ e t z 1981; , Petz and Zembnek, 19881. We conclude the section by stating a recent characterization of traces [Hamhalter, 20041 which concerns overall dispersion. Let p be a probability measure on the projection lattice P ( M ) . The overall dispersion, a(p), is defined as U(P) = SUP lib) - P ~ J ) ~ . PEP(M)

Let us observe that the overall dispersion is zero exactly when the measure is dispersion free. Given any nonzero positive finite measure p on P ( M ) we define the overall dispersion a(p) as that of the normalized measure - A measure ~ ( 1' ) p : P ( M ) + [0,oo] is said to have the minimal dispersion property if for each subalgebra N of M which is isomorphic to M3(@)and for which p is a finite and nonzero measure on P ( N ) , p has the smallest dispersion on P ( N ) among all states on P ( N ) . A measure p is said to be join-dense if for each nonzero projection e in P ( M ) there is a nonzero projection f 5 e such that p(f) < oo. The following characterization of traces has been obtained in [Hamhalter, 20041. THEOREM 52. Let M be a von Neumann algebra with zero abelian direct summand and without Type I2 direct summand. A semifinite normal weight p on M i s a trace if, and only if, pIP(M) i s join-dense and has the minimal dispersion property.

4.2

Lattice structure and a-additivity

The central topic of this paragraph is the relationship between a-additivity of noncommutative probability measures and their lattice-theoretic properties. The main property we consider, called today the Jauch-Piron property, was introduced by M. Jauch and C. Piron in the 60's in connection with the propositional calculus of quantum mechanics and the problem of hidden variables [Jauch and Piron, 1963; Jauch and Piron, 1969; Jauch, 19681. Let p be a state (i.e. a probability measure) on an orthomodular lattice, L. We say that p is a Jauch-Piron state if the following

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condition holds: If p(a) = p(b) = 0 for a, b E L, then p(a V b) = 0. A physical motivation for introducing this concept was the folowing. The elements of L are interpreted as propositions (yes-no observables or experiments) on the quantum system in a state p. The value p(a), (a E L), represents the probability that the proposition a is true. The Jauch-Piron property now means: "if the proposition a is false and the proposition b is false then the proposition (a or b) is also false. From the probabilistic point of view, the Jauch-Piron property encodes the following largely adopted rule of Kolmogorovian probability theory: If two events A and B have zero probability, then the probability of occurring A or B is also zero. Another motivation for investigating Jauch-Piron states is purely mathematical. If e is a state on a von Neumann algebra M, then its left kernel L, = {x E M I e(x* x) = 0) is a norm closed left ideal of M . The natural question arises of what are the properties of states e for which L,flP(M) is a lattice ideal in the projection lattice P(M). Such states show a nice interplay with the projection lattice structure. Let us recall that a set I C P(M) is an ideal in P ( M ) if, for all projection p, q E P(M), the following conditions hold true: (i) p V q E I whenever p, q E I (ii) if p 5 q and q E I, then p E I. The condition (ii) is automatically satisfied for every set L, fl P ( M ) by the monotonicity. Hence, we see that the Jauch-Piron states are precisely the states that generate lattice ideals in the von Neumann projection lattices. If M is abelian then any state on M is obviously Jauch-Piron. The converse holds only if the nonabelian part of the algebra is finitedimensional. Let us remark that subadditive states treated in the previous part are obviously Jauch-Piron. A thorough analysis of continuity properties of Jauch-Piron states has been carried out by L. J. Bunce and J. Hamhalter in a series of papers [ ~ u n c e and Hamhalter, 1994; Bunce and Hamhalter, 1995; Bunce and Hamhalter, 1996; Bunce and Hamhalter, 2000; Hamhalter, 1993b], for a survey see [Hamhalter, 2003, Chapter lo]. This program has shown that there is a surprisingly strong interplay between the continuity property of Jauch-Piron states (nonsingularity, Kolmogorovian regularity), the type of the algebra in question, and the size of its centre. For example, it has been shown that any Jauch-Piron state on a factor is regular (i.e. its kernel has to be closed under countable unions of projections). As a starting point of our discussion, let us remark that Jauch-Piron state may be far from being a-additive. For example, any tracial state (and so any state on a commutative algebra) is Jauch-Piron, however any von Neumann algebra with infinite-dimensional centre admits a singular tracial state. From another point of view, any faithful state is Jauch-Piron without being necessarily a-additive. However, it turns out that a-additivity can be naturally described in terms of the Jauch-Piron property if we consider the collection of all transformed states. Let Q be a state on a von Neumann algebra M . Given a E M such that e(a* a) > 0, the transformed state, e,, is defined by

Let E, denote the norm closure of the set

{ea I @(a*a) # 0, a

E M). It can be

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verified that E, is the set of all states whore, where h is a unit vector in the G.N.S. Hilbert space He (see e.g. [Kadison, 19621). From the point of view of physics, the transformed states correspond to manipulations with the system initially prepared in the state p. THEOREM 53. Let M be a v o n N e u m a n n algebra without a n abelian direct summand. A state p o n M is a-additive if, and only if, all states in Ee are JauchPiron. The previous result says that if the entire collection of states that can be approximated by the transformations of a given state p consists of Jauch-Piron states, then a-additivity of p follows automatically. The reversible dynamics of a closed quantum mechanical system is given by the group of unitary operators. On the other hand, when the system is coupled to an environment or subject to a measurement, its most general time evolution is irreversible, which means that it does not preserve all structural properties. Mathematically, it is given by a normal completely positive map. By Kraus's representation theorem [ ~ r a u s19831 , normal completely positive maps (called quantum operations) are in important cases direct sums of the transformations x -t a* x u considered above. In this light, we can slightly reformulate the preceding theorem by saying that a state p is a) Jauch-Piron for each suitable quantum operation additive if, and only if, ~ ( 7 . is 7. If p is a pure state, then every state in Ee is unitarily equivalent to p. Since the Jauch-Piron property is preserved by the unitary transforms we have the following corollary: COROLLARY 54. Let M be a v o n N e u m a n n algebra without nonzero abelian direct summand. A pure state p o n M i s a-additive iJ and only if, p k Jauch-Piron. Under certain circumstances this automatic continuity holds for factor states. (A factor state is a state whose image in the G.N.S. representation generates a factor.) For example if M is a properly infinite a-finite von Neumann algebra, then any factor Jauch-Piron state has to be a-additive (for more details see [Bunce and Hamhalter, 2000]). It is obvious from our discussion that a-additivity emerges when the given state has nice lattice-theoretic properties. Another striking result along this line tells us that all lattice homomorphisms are a-additive in the noncommutative case. THEOREM 55. Let T : M -+ N be a *-homomorphism between v o n N e u m a n n algebras, where M has zero abelian part. T h e n T i s a-additive if, and only iJ for all projections e, f E M, ~ ( Vef ) = ~ ( eV) ~ ( f .) As a result, a state p on an essentially noncommutative algebra is a-additive exactly when its G.N.S. representation is a lattice homomorphism. Let us mention a few open problems in this area. It is not known whether all factor Jauch-Piron states on properly infinite von Neumann algebras are a additive. A positive answer has been obtained only under the assumption of the

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continuum hypothesis [ ~ u n c eand Hamhalter, 20001. Moreover, it follows from hitherto known results on the geometric structure of the Jauch-Piron state space that Jauch-Piron states are norm dense in the state space of any a-finite von Neumann algebra. Whether the same holds for all von Neumann algebras remains an open question. There are also Jordan versions of the above results based on the fact that a Jauch-Piron state on a J W algebra W extends usually to a Jauch-Piron state on a von Neumann algebra generated by W. This analysis may be found in [ ~ u n c e and Hamhalter, 19961. In summary, the requirement of a-additivity of a probability measure in a classical Kolmogorovian probability space (which is necessary e.g. for the onesided continuity of the distribution function) is an indispensable technical axiom that cannot be derived from the order structure of events. In contrast to this, in "proper" quantum measure theory one need not postulate a-additivity because it is already a consequence of natural lattice-theoretic axioms involving only finitely many operations in the projection lattice. In this respect the axiomatics of quantum theory may, paradoxically, seem to be simpler. ACKNOWLEDGMENT The work was supported by the research plan of the Ministry of Education of the Czech Republic No. 6840770010. Besides, the author acknowledges the support of the Grant Agency of the Czech Republic, Grant. no. 201/03/0544, "Noncommutative Measure Theory" and the support of the Alexander von Humboldt Foundation, Bonn, Germany. BIBLIOGRAPHY [Aarnes, 19661 J. F.Aarnes. The Vitali-Hahn-Saks Thwrem for von Neunzann algebras, Math. Scand. 1 8 , 87-92, 1966. [Aarnes, 19691 J. F.Aarnes. Physical states on a C8-algebra, Acta Math. 122, 161-172, 1969. [Aarnes, 19701 J. F. Aarnes. Quasi-states on CY-algebras,mans. Amer. Math. Soc., 149, 601625, 1970.

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[Reedman, 19971 W. Reedman. A n alternative Dunford-Pettis property, Studia Mathematica, 125, 143-159, 1997. [Gell-Mann and Hartle, 1990a] M. Gell-Mann and J. Hartle. Quantunz mechanics i n the light of quantum cosmology, Volume VIII, Complexity, Entropy and the Physics of Information, pp. 425458. SF1 Studies in the Science of Complexity, Addison-Wesley, Reading, 1990. [Gell-Mann and Hartle, 1990bl M. Gell-Mann and J. Hartle. Alternative dewhering histories in quantum mechanics, Proc. 25th Int. Conf. on High Energy Physics, August, 2-8, (1990), World Scientific, Singapore, 1990. [Gleason, 19571 A. M. Gleason. Measures on the closed subspaces of a Hilbert space, Journal of Mathematics and Mechanics, Vol. 6, No. 6, 885-893, 1957. [Graves and Reuss, 1980) W. H. Graves and W. Ruess. Compactness i n spaces of vector-valued measures and a natural topology for spaces of bounded measurable functions, Contemp. Math. 2, 189-203, 1980. [Grothendieck, 19531 A. Grothendieck. Sur les applications lindaira faiblenaent compactes d'espaces du type C ( K ) , Canadian J. Math. 5, 129-173, 1953. [Gudder, 1979a] S. P. Gudder. Stochastic Methods i n Quantunz Mechanics, Elsevier, North Holland, 1979. [Gudder, 1979b] S. P. Gudder. A Radon-Nikodynz theorem for *-algebras, Pacific Journal of Mathematics, Vol. 80, No. 1, 141-149, 1979. [Gunson, 19721 J. Gunson. Physical states on quantum logics I, Ann. Inst. Henri Poincar6, Vol. XVII,No. 4, 295-311, 1972. [Haag, 19921 R. Haag. Local Quantunz Physics, Fields, Particles, Algebras, Texts and Monographs in Physics, Springer Verlag, 1992. [Hamana, 19771 M. Hamana. O n linear topological properties of some C*-algebras, Tohoku Math. Journ. 29, 157-163, 1977. [Hamhalter and P t i k , 19871 J. Hamhalter and P. Ptik. A completeness criterion for inner product spaces, Bull. London Math. Soc. 19, 259-263, 1987. [Hamhalter, 19921 J. Hamhalter. Additivity of vector Gleason measures, Int. J. Theor. Physics, 31, No. 1, 1-13, 1992. [Hamhalter, 1993a] J. Hamhalter. Centrally detemzined states on von Neumann algebras, Mathematica Bohemica, Vol. 117, No. 2, 1889-1892, 1993. [Hamhalter, 1993bj J. Hamhalter. Pure Jauch-Piron states on von Neumann algebras, Ann. Inst. H.Poincar6, Phys. T h b r . 58, 173-187, 1993. [Hamhalter, 19941 J. Hamhalter. Gleason property and eztensions of states on projection logics, Bull London Math. Soc. 26, 367-372, 1994. [Hamhalter, 1997a] J. Hamhater. The Nikody'm Boundedness Theorem, Tatra Mountains Math. Publ. 1 0 , 179-181, 1997. [Hamhalter, 1997b] J. Hamhalter. Statistical independence of operator algebras, Ann. Inst. Henri Poincar6 , Vol. 67, no. 4, 447-462, 1997. [Hamhalter, 1999a] J. Hamhalter. Universal state space embeddability of Jordan-Banach algebras, Proc. Amer. Math. Soc., Vol. 127, Num 1, 131-137, 1999. [Hamhalter, 1999b1 J. Hamhalter. Supporting sequences of pure states on J B algebras, Studia Mathematica 136, (I), 3 7 4 7 , 1999. [Hamhalter, 2002aj J. Hamhalter. Multiplicativity of eztremal positive maps on abelian parts of operator algebras, J. Operator Theory, 4 8 369-383, 2002. [Hamhalter, 2002bl J. Hamhalter. C*-independence and W*-independence of von Neumann algebras, Mathematische Nachrichten, 2 3 4 2 4 0 , 146-156, 2002. [Hamhalter, 2002~1J. Hamhalter. Nonwmmutative phenomena in measure theory on operator algebras, findiconti DelllUniversita Di Trieste, Nuova Serie, Rend. Inst. Mat. Univ. 'Ikieste Vol. XXXIV, 19-43, 2002. [Hamhalter, 20031 J. Hamhalter. Quantum Measure Theory, Kluwer Academic Publishers, Dordrecht, Boston, London, 2003. [Hamhalter, 20041 J. Hamhalter. Traces, dispersions of states and hidden variables, Foundations of Physics, Letters, Bol. 17, Nu. 6, 581-597, Issue 10, 2101-2111, 2004. [Hamhalter, 20051 J . Hamhalter. States o n Operator Algebras and Axionaatic System of Quant u m Theory, International Journal of Theoretical Physics, Vol. 44, No. 11, 1941-1954, 2005. [Hanche-Olsen and Stormer, 19841 H. Hanche-Olsen and E. Stormer. Jordan Operator Algebras, Pitman Publishing, Boston, London, Melbourne, 1984.

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[Isham, 19941 C. J. Isham. Quantum logic and histories approach to quantum theory, J.Math.Phys. 35, (5), 1994. [Isham et al., 19941 C. I. Isham, N. Linden and S. Schreckenberg. The classification of decoherence functionals: A n analog of Gleason's theorem, J. Math. Phys. 35, (12), 1994. [Isham and Linden, 1997) C. I. Isham and N. Linden. Infornzation entropy and the space of decoherence functions i n generalized quantum theory, Physical Review A, Vol. 55, Nu. 6, 4030-4040, 1997. [Jauch and Piron, 19631 M. Jauch and C. Piron. Can hidden variables be excluded i n quantum mechanics ?, Helvetica Physica Acta, 36, 827-837, 1963. [Jauch and Piron, 19691 M. Jauch and C. Piron. O n the structure of quantum proposition systems, Helv. Phys. Acta 42, 827-837, 1969. [Jauch, 19681 M. Jauch. Foundations of Quantum Mechanics, Reading, MA, Addison-Wesley, 1968. [Kadison, 19551 R. V. Kadison. O n the additivity of the trace i n finite factors, Proc. Nat. Acad. Sci. U.S.A. 41, 385-387, 1955. [Kadison, 19621 R. V. Kadison. States and representations, Trans. Amer. math. Soc. 103, 304319, 1962. [Kadison and Ringrose, 19861 R. V. Kadison and J. R. Ringrose. Fundamentals of the Theory of Operator Algebras, Academic Press, Vol. I, 11, 111, IV,1986. [Kalmbach, 19831 G. Kalmbach. Orthomodular lattices, Academic Press, Inc., 1983. [Kochen and Specker, 19671 S. Kochen and E. P. Specker. The problem of hidden variables i n quantum mechanics, J. Math. Mech. 17, 59-87, 1967. [Kraus, 19831 K. Kraus . States, Effects and Operations: Fundamental Notions of Quantum Theory, Ledure Notes in Physics, Vol. 190, Berlin, Springer-Verlag, 1983. [Mackey, 19631 G. W. Mackey. The Mathematical Foundations of Quantum Mechanics, New York, Benjamin, 1963. [Maeda, 199O] S. Maeda. Probability measures o n projections i n von Neumann algebras, Reviews in Mathematical Physics, 1,235-290, 1990. [Matvejchuk, 1997a1 M. Matvejchuk. Semiwnstant measures o n hyperbolic logics, Proc. Amer. Math. Soc., Vol. 125, Nu 1, 245-250, 1997. [Matvejchuk, 1997b] M. Matvejchuk. Gleason's theorem i n space with indefinite metric, Math. Nachr. 184, 229-243, 1997. [Matvejchuk, 19811 M. Matvejchuk. Description of finite measures in semifinite algebras, Vol. 15, 41-53, 1981. [ ~ a t v e j c h u k1998a1 , M. Matvejchuk. Several methods for solving the Gleason problem, Measures on Projections and Orthomodular Posets, Kazan Mathematical Society, 14-37, 1998. [Matvejchuk, 1998bl M. Matvejchuk. Probability measures i n W * J-algebras i n Hilbert spaces with conjugation, Proc. Amer. Math. Soc. Vol. 126, Nu. 4, 1115-1164, 1998. [, ] A.Mayet-Ippolito: Generalized orthomodular posets, Demonstration Mathematica, Vol. XXIV, NO. 1-2, (1991), 263-274. [Mermin, 19931 N. D. Mermin. Hidden variables and two theorems of John Bell, Rev. Mod. Phys. 65, 803-815, 1993. [Misra, 19671 B. Misra. When can hidden variables be excluded i n quantum mechanics?, Nuovo Cimento 47A, 841-859, 1967. [Murphy, 19901 G. J. Murphy. C*-algebras and Operator theory, Academic Press, Inc., 1990. [Murray and von Neumann, 19361 F. J. Murray and J. von Neumann: O n rings of operators, Ann. Math., Vol. 37, 116-229, 1936. [Mushtari, 19891 D. Mushtari. Logics of projectors i n Banach spaces, Izv. Vysch. Uchebn., Zaved. Mat., Vol. 33, No. 8, 44-52, 1989. [Navara, 19881 M. Navara. Quantum logics with the Radon Nikodym property, Order 4,387-395, 1988. [von Neurnann, 19551 J. von Neumann. Mathematical Foundations of Quantum Mechanics, Princeton, New Jersey: Princeton Univ. Press, 1955. [Nishirnura, 19841 H. N i s h i i a . A Boolean-valued approach to Gleason's theorem, Institute of Mathematics, University of Tskuba, Ibaraki, Japan, 1984. [Parthasarathy, 19921 K. R. Parthasarathy. A n Introduction to Quantuna Stochastic Calculus, Birkhauser Verlag, Basel, 1992. [Paszkiewicz, 19851 A. Paszkiewicz. Measures o n projections in W*-factors, J. Fund. Anal. 62, 87-117, 1985.

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[Pfitzner, 19941 H. Pfitzner. Weak compactness in the dual of a C*-algebra is determined commutatively, Math. Ann. 298, 349-371, 1994. [~edersenand Takesaki, 1973) G. K. Pedersen and M. Takesaki. The Radon-Nikodym theorem for von Neumann algebras, A d a Math. 130, 53-87, 1973. [Pedersen, 19791 G. K. Pedersen. C*-Algebras and their Automorphisms Group, New York, London, Academic Press, 1979. [Pelczynski, 19621 A. Pelczynski. Banach spaces on which every unconditionally converging operator is weakly compact, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom Phys. 10, 641-648, 1962. [ ~ e r e s 19911 , A. Peres. Two simple proofs of Kocher-Specker theorem, J. Phys. Math. Rev. Gen. A 24, 175-178, 1991. [Peres, 19951 A. Peres. Quantum Theory, Concepts and Methods,, Kluwer Academic Publishers, Dordrecht, 1995. [ ~ e t z19811 , D. Petz. A characterization of the canonical centre-valued trace in finite von Neunzann algebras, J. London. Math. Soc. 2, 329-331, 1981. [Petz and Z e d n e k , 19881 D. Pets and J. Zemdnek. Characterization of the trace, Linear Algebra and its Applications, 111,43-52, 1988. [Piron, 19641 C. Piron. Axiomatique quantique, Helv. Phys. Acta 37, 439-468, 1964. iron, 19681 C. Piron. Foundations of Quantum Mechanics, Addison-Wesley Publishing Company, Reading, Massachusetts, 1968. [pitowski, 19981 I. Pitowski. Infinite and finite Gleason's theorems and the logic of indeternzinacy, J. Math. Phys. 39, 1998, 218-228. [Plymen, 19681 R. J. Plymen. Dispersion free normal states, I1 Nuovo Cimento, Vol. LIV A, N. 4, , 862-870, 1968. [Ptik and Pulmannovd, 19911 P. Pt& and S. Pulmannovd. Orthomodular Structures as Quantum Logics, Academic Publishers, Dordrecht, (Boston), London, 1991. [Pt&, 19851 P. Ptdk. On extensions of states on logics, Bull. Polish Acad. Sci. Math. 33, 493497, 1985. [Redei, 19981 M. Redei. Quantum Logic in Algebraic Approach, Kluwer Academic Publishers, Fundamental Theories of Physics, Dordrecht, Boston, London, 1998. [ ~ u d ~and i ~ Wright, h 19981 0. Rudolph and J. D. M. Wright. The multi-form generalized Gleason theorem, Commun. Math. Phys. 198, 705-709, 1998. [ ~ u d o l and ~ h Wright, 19991 0. Rudolph and J. D. M. Wright. Homogeneous dewherence functionals i n standard and history quantum mechanics, Commun. Math. Phys. 204, 249-267, 1999. [Rudolph, 20001 0. Rudolph. The representation theory of dewherence fvnctionals in history quantum theories, International Journal of Theoretical Physics, Vol. 39, No. 3, 871-884, 2000. [Rudolph and Wright, 20001 0. Rudolph and J. D. M. Wright. On unentangled Gleason theorems for quantum information theory, Letters in Mathematical Physics, 52, 239-245, 2000. [Riitimann, 19971 G. Riitimann. Jauch-Piron states, J. Math. Phys. 18 (2), 189-193, 1997. [Sakai, 19711 S. Sakai. C*-algebras and W*-algebras, Ergebnisse der Mathematik 60, SpringerVerlag, Berlin, Heidelberg, New York, 1971. [Sakai, 19651 S. Sakai. A Radon-Nikodyrn theorem in W*-algebras, Bull. Amer. Math. Soc. 71, 149-151, 1965. [Sakai, 19911 S. Sakai. Operator Algebras in Dynantical Systems, Encyklopedia of Mathematics and its application 41, Cambridge University Press, 1991. [Segal, 19531 I. E. Segal. A non-commutative extension of abstract integration, Ann. Math. 57, 401-457, 1953. [Stolyarov, 19921 A. I. Stolyarov and 0. E. Tikhonov. On a characterization of traces in terms of the non-wmnzutative integration, VINITI, 1,3186-B96, 1992. (in Russian). [Stratilla and Zsido, 19791 S. Stratilla and L. Zsido. Lectures on von Neuntann Algebras, Tunbridge Wells: Abacus Press, 1979. [~akesaki,19791 M. Takesaki. Theory of Operator Algebras I, Berlin, Heidelberg, New York, Springer, 1979. [~akesaki,20021 M. Takesaki. Theory of Operator Algebras II, III, Berlin, Heidelberg, New York, Springer, 2002. ishe her, 19821 J. Tisher. Gleason's Theorem for '&pe I uon Neunzann algebras, Pacific J. Math. 100, 273-288, 1982.

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[Tisher, 19791 J. Tisher. A note on the Radon-Nikodym theorem of Pedersen and Takesaki, A d a Sci. Math (Szeged), 41, no. 3-4, 411-418, 1979. [Topping, 19651 D. Topping. Jordan Algebras of Self-Adjoint Operators, Memoirs of the Amer. Math. Soc., 53, 1965. [Varadarajan, 19681 V. S. Varadarajan. Geometry of Quantum Theory, Vol. I , Van Nostrand, Princeton, 1968. [Wallach, 20021 N. R. Wallach. An unentangled Gleason's theorem, Contemporary Mathematics, Vol. 305, 291-298, 2002. [Wilbur, 19751 W. J. Wilbur. Quantum logic and locally convez spaces, Transactions of the American Mathematical Society, 207, 343-360, 1975. [Wilbur, 19771 W. J. Wilbur. O n characterizing the standard quantum logics, Transactions of the American Mathematical Society, 233, 265-282, 1977. right, 19951 J. D. M. Wright. The structure of dewherence functionals for von Neumann quantum histories, J . Math. Phys. 36, (lo), 5409-5431, 1995. [Wright, 19981 J. D. M. Wright. Dewherence functionals for von Neumann quantum histories: Boundedness and countable additivity, Comrnun. Math. Phys. 191, 493-500, 1998. [Yeadon, 19831 F. J. Yeadon. Measures on projections i n W *-algebras of Type 1 1 1 , Bull. London Math. Soc. 15, 139-145, 1983. [Yeadon, 19841 F. J. Yeadon. Finitely additive measures on projections i n finite W*-algebras, Bull. London Math. Soc., 16, 145-150, 1984.

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HANDBOOK O F QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM STRUCTURES Edited by K. Engesser, D. M. Gabbay and D. Lehmann O 2007 Elsevier B.V. All rights reserved

335

CONSTRUCTIONS OF QUANTUM STRUCTURES Mirko Navara

1 INTRODUCTION

By quantum structures we mean systems of events modelling abstract quantum mechanical experiments. They are equipped with the corresponding relations and operations derived from their interpretation. Depending on the assumptions, we obtain different levels of generality, from orthomodular lattices and orthomodular posets to orthoalgebras and effect algebras. We use the term quantum structure for any of these, depending on the context (which will always be specified). We describe common features of these structures, as well as their particular properties. We leave the possible interpretations to physicists; they should choose an adequate model which describes optimally the real systems of quantum mechanics. Here we study the mathematical properties which might decide the adequacy of particular models. This chapter is devoted to constructions of quantum structures. Particular attention is paid to pastings which seem to be special and typical of this field and which make it quite different from the theory of Boolean algebras or MV-algebras. They are essential for generating examples which disprove various conjectures. Beside this, they also contributed numerous positive results showing what is possible in this area. No rich mathematical theory can emerge without such constructions. The contemporary techniques have not been published together and in a unified form so far. This is the main aim of this chapter. Besides, we enrich it by several tools that have not been published yet. i j PulmannovB, 20001 or Our notation mostly follows that of [ ~ v u r e ~ e n s kand [Ptik and Pulmannovi, 19911. 2 QUANTUM STRUCTURES Here we define the basic structures treated in this chapter and we clarify their mutual relations. Although the most general quantum structures studied here are effect algebras, we often restrict our attention to orthoalgebras which seem to be easier to understand. More specific quantum structures, orthomodular posets and lattices, are introduced as special types of orthoalgebras. For the

Mirko Navara

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standard approach to orthomodular posets and lattices, we refer to [Beltrametti and Cassinelli, 1981; Beran, 1984; Kalmbach, 1983; P t B and PulmannovA, 1991; Varadarajan, 19681.

2.1

Efect algebras and orthoalgebras

DEFINITION 1. An effect algebra (EA) is a quadruplet (L, @, 0, I ) , where L is a set, 0 , l E L, and @ : L2 + L is a partial binary operation (orthosum) satisfying the following properties for all a, b, c E L:

(EA3)

For each a E L, there is exactly one d E L such that a @ d = 1.

(EA4)

a @ 1is defined iff a = 0.

An orthoalgebra (OA) is an effect algebra satisfying the following strengthening of (EA4): (OA)

a @ a is defined iff a = 0.

REMARK 2. When we use partial operations (incl. @) in expressions, we a u t e matically assume their existence. Thus (EA1) and (EA2) should be understood as follows: If one side of the equality exists, then the other exists, too, and both sides are equal. Sometimes we speak briefly of an effect algebra L instead of (L, @, 0 , l ) . When there is a risk of confusion, we index the operations by the corresponding algebra, e.g., @L, OL, 1 ~ . EXAMPLE 3. Every Boolean algebra becomes an orthoalgebra if we define the orthosum by a @ b = a V b whenever a A b = 0. EXAMPLE 4. Let us take L = (0, a, 1) and define a @ a = 1,x @ 0 = x for all x 1: L. Then (L, @, 0 , l ) is an effect algebra which is not an orthoalgebra. EXAMPLE 5 . . Let H be a Hilbert space. We denote by 0 the zero operator and by 1the identity on H. Let & ( H )be the set of all self-adjoint operators Q on H such that 0 5 Q 5 1 and let P ( H ) C &(H) be the set of all projectors on H, i.e., P ( H ) = {P E &(H) I P2 = P). If P, Q E E(H), we define P @Q = P Q if P Q 5 1 . Then (&(H),@, 0 , l ) and (P(H),@, 0 , l ) are effect algebras; (P(H), @, 0 , l ) is also an orthoalgebra.

+

+

EXAMPLE 6. Let (R, +, .,0 , l ) be a ring with unity and let I(R) be the set of all its idempotents. Let us define a @ b iff a - b = b . a = 0, in this case we put a @ b = a + b. Then (I(R), @, 0 , l ) becomes an orthoalgebra.

Constructions of Quantum Structures

337

DEFINITION 7. For a poset (P, 5 ) and a, b E P , a 5 b, we define the interval P[a, b] = {c E P I a 5 c 5 b). An element a of a bounded poset (P, consequence of (OA), an effect algebra is an orthoalgebra iff it csntains no isotropic elements.

2.2

Quantum structures as posets

For an effect algebra (L, @, 0 , l ) and a, b E L, we define (Ord)

a 5 b iff there is an element c E L such that b = a @ c.

Then 5 is a partial order inducing partial lattice operations A, V on L. By an atom of an effect algebra we mean an atom of the corresponding poset. DEFINITION 9. If an effect algebra is a lattice w.r.t. the ordering defined by (Ord), it is called a lattice effect algebra (LEA). An effect algebra is called chainfinite if all chains of the corresponding poset are finite. We define a unary operation ': L 4 L assigning to each a E L the unique element d = a' satisfying (OA3). It is called the orthosupplement. REMARK 10. In early papers on effect algebras the orthosupplement was called the orthocomplement. This term is historically used for orthoalgebras and more

Mirko Navara

338

specific structures. The different term "orthosupplement" emphasizes the fact that some properties typical for orthocomplements are lost in effect algebras. If L is an orthoalgebra, the ordering and the orthosupplementation induce the structure of an orthoposet, defined as follows: DEFINITION 11. [ ~ e r a n1984; , Kalmbach, 19831 An orthoposet (OP) is a pentuplet (L, 0,1, '), where (L, 0, then 8 is the usual difference of real functions ( 9 8 f ) ( t ) =g(t) - f ( t ) and if @(x) = x2 then

A function Q, of the above kind is called a generator of a difference. Moreover, if @(I) = 1, then @ is called a normed generator. We refer to Mesiar [1995],Mesiar and Pap [1994] for more information on some other properties of a difference operation of fuzzy sets. DEFINITION 5. Let T be a D-a-poset of fuzzy sets. A probability measure (a state) on T is a mapping rn : T + [0, 11 such that

Ferdinand Chovanec and FtantiEiek Kdpka

372

(P2) If ( f n ) n € N C T , fn 5 fn+l and

V

fn = f ,

then

nEN

EXAMPLE 6 . Let T [0,1IX be a D-poset of fuzzy sets. Let to E X such that f ( t o ) exists for every f E T . Then the mapping s : T + [O,1]defined by

is a state on T . DEFINITION 7 . Let F be a D-a-poset of fuzzy sets. Let B ( R ) be a Borel aalgebra of the real line R . The mapping x : B ( R ) + 3 is said t o be an observable on T , if the following conditions are satisfied:

( 0 2 ) If (An)nENis a sequence of Borel sets such that An then x(An) 5 x(An+l), and x

( U An) nEN

(03) If A , B are Borel sets, A

=

V

An+1 for every n E N ,

x(An).

nEN

B , then x ( B \ A) = x ( B ) 8 x ( A ) .

It is easy to prove that if x : B ( R ) + T is an observable on T ,then the mapping m , : B ( R ) + [O, 11, m , ( E ) = m ( x ( E ) ) is a probability measure on B ( R ) . The mapping m , is said to be a probability distribution of the observable x in the state m. Now the mean value of the observable x in the state m can be defined by the integral

if it exists and is finite. Many results concerning questions of the probability theory on D-posets of fuzzy sets can be found in Chovanec and JureEkovB [1992],Chovanec and K6pka [1992],Chovanec and E. RybBrikovB [1997],KGpka [1995a],Mesiar [1994],RieEan [1996;20001. Topological properties of D-posets of fuzzy sets were studied in Palko [1995; 19971.

We can generalize the concept of a D-poset of fuzzy sets to an abstract partially ordered set, where the basic operation is difference. We thus arrive at a very

general and, at the same time, very simple structure - a D-poset [Kbpka and Chovanec, 19941. First we define a partial difference operation on partially ordered sets. DEFINITION 8. Let (P, 5) be a non-empty partially ordered set (a poset). A partial binary operation 8 is said to be a difference on P, if an element b 8a is defined in P if and only if a 5 b, and the following conditions are satisfied. (Dl) b a a

5 b.

(D2) b e ( b @ a ) = a . (D3) I f a s b s c , t h e n c 9 b s c g a a n d ( c @ a ) 8 ( c 8 b ) = b 8 a .

A structure (P,

2, 8 ) is called a poset with difference.

For simplicity of notation, we will write P instead of (P, 5 , 8 ) . Obviously, the set-theoretical difference of subsets of a non-empty set and the usual difference of non-negative real numbers satisfy conditions (Dl) - (D3). Moreover, if C is an orthomodular poset then the operation 9 defined by b 8 a = b A a' whenever a

5 b , (a' is the orthocomplement of a),

is a difference on L. PROPOSITION 9. Let P be a poset with diflerence and a, b, c, d E P. The following assertions are true. (i)

Ifasbsc7thenb8a5c8aand(c8a)8(b8a)=c8b.

(ii) I f b s c a n d a S c g b , t h e n b s c 8 a a n d ( c 8 b ) 8 a = ( c ~ a ) 8 b . (iii) If a

5 b 5 c,

then a 5 c 9 (b 9 a) and (c 8 (b 8 a)) 8 a = c 8 b.

(iv) I f a ~ c a n d b ~ c , t h e n c 8 a = c 8 b i f a n d o n l y i f a = b . (v) I f d s a s c , d s b s c , t h e n c e a = b 8 d i f a n d o n l y z f c 8 b = a 8 d d . (vi) a 9 a is the minimal element in P. Proof. (i) From (Dl) and (D4) we get that (c 9 a) 9 (c 9 b) = b 9 a

(ii) From the assumptions it follows that a

2 c 9 a and

5 c 8 b 5 c, and from (D4) we obtain

Because, by (i), ( c 9 b ) e a 5 tea, we obtain that ( c e a ) 8 ( c e b ) 8 a = ce(c8b) = b, therefore (C9 a) 9 b = (c 8 a) 8 ((c 8 a) 9 ((c 8 b) 8 a)) = (c 8 b) 8 a .

374

Ferdinand Chovanec and Fkantisek KBpka

(iii) By (i) we have b 8 a

5 c 9 a 5 c, and by (D4) we obtain a = c 8 ( c 8 a) 5 c 9 ( b e a).

Using (ii) and (i), we obtain

(iv)Ifc8a=ceb,thenb=ce(c8b)=c8(~8a)=a. The converse assertion is obvious. (v) If c 8 a = b e d , then

The converse assertion can be proved analogously. (vi) Let w E PI w 5 a 9 a . Then w 5 a and

hence, a 8 w = a. Therefore, a 9 a

=a8

( a 8 w) = w.

1

PROPOSITION 10. Let P be a poset with difference, and let {at : t E T ) be a system of elements of P and c E P such that at 5 c for any t E T,where T i s a n index set. If the least upper bound V at exists in P,then the greatest lower bound t--FT A ( c 9 a t ) exists in P , and tET

A (c 8 at)

If P i s a lattice, then from the existence of

the existence of

tET

V

at

tET

follows, and the above equality holds.

Proof. The inequality at 5

V LET

at gives c 8

V

at 5 c 8 at for any t E T. If w E P

tET

is an element such that w 5 c 8 at for any t E T , then at 5 c 8 w for any t E T . Therefore V at 5 c e w and hence w 5 c e V at, which implies that the element

c9

V

tET

tET

at is the greatest lower bound of the set { c 8 at : t E T ) .

tET

We proceed similarly for the second part of the assertions.

1

We now introduce the concept of a lower bounded poset with difference. PROPOSITION 11. Let P be a poset with difference and with least element Op. T h e n the following assertions are true. (i)

a 8 O p = a for all a E

P.

(ii) a 8 a = Op for all a E P . (iii) I f a , b ~ P , a s b t, h e n b 8 a = O ~ i f a n d o n l y i f b = a .

(iv) I f a , b € P , a s b , t h e n b 8 a = b i f a n d o n l y i f a = O p . (v) If a V b E P, then ( ( a V b) 8 a ) A ( ( aV b) 8 b) E P and ( ( a V b) 8 a ) A ( ( a V b) 8 b) = Op. Proof. (i) For every a E obtain

P we have Op 5 a 8 a 5 a. From (D2) and (D3) we a=a8(a8a)sa80psa

which implies a 8 O p = a. (ii)From the above and (D2) we have a 8 a = a 8 ( a 8 O p ) = Op. T h e proofs o f (iii)and ( i v ) are obvious. ( v )It suffices t o put c = a V b in Proposition 10. Since algebraic structures relevant t o probability theory must contain a greatest element, we define the structure o f a D-poset as follows. A poset P with difference containing a greatest element l p , and consequently, a smallest element Op = l p 8 l p , is called a diflerence poset ( a D-poset, i n short). B y a D-a-poset we mean a D-poset P satisfying the following condition.

(D4)I f (an)nENG P , a, 5 an+l for any n E N , then

V

nEN

a, E P .

T h e following theorem gives rise t o an alternative definition o f a D-poset. T H E O R E M 12. ( [ ~ a v a r aand Ptcik, 19971) Let P be a bounded partially ordered set with the least element Op and the greatest one l p . Let 8 be a partial binary difference operation on P such that the element b 8 a exists in P i f and only i f a 5 b. Then P is a D-poset i f and only if the following conditions hold. ( 1 ) a 8 O p = a f o r any a € P .

For any element a in a D-poset P , the element 1p 8 a is called t h e orthosupplement o f a and is denoted b y aL. T h e unary operation Iis an involution and an order reversing operation on P, i.e., a l l = a , and, if a 2 b then bL 5 aL. PROPOSITION 13. Let {at}tET be a system of elements of a D-poset P such that /\ at E P and let c E P , c 5 at for any t E T . Then tET

Ferdinand Chovanec and Frantisek KBpka

376

Proof.

In a D-poset, we can define an operation (orthosummation) @ dual t o the operation €3 as follows. a 63 b = (aL 8 b ) l , for a

5 bL.

I t is easy t o see that the operation @ has the following properties. (a)

is a commutative operation (in the sense such that if a @ b E P , then b@a~Panda@b=baa). @

(b) @ is an associative operation (in the sense such that if b @ c E P and a @ ( b @ c ) ~ P , t h e n a @ b E P( a, @ b ) @ c E P a n d a @ ( b @ c ) = ( a @ b ) @ c .

(c) a @ aL = lF for every a E P . ( d ) a @ O p = a for every a E P. T h e relationship between the operations 8 and @ is the topic o f the next proposition. PROPOSITION 14. Let P be a D-poset. Then the following assertions are true for every a, b, c E P .

(i) a

2 bL

implies a

5 ( a @ b) and

( a 63 b) €3 a = b.

(ii) a s b L and ( a @ b )s c i m p l y c 8 ( a @ b ) = ( c 8 a ) 8 b = ( c 8 b ) 8 a .

5 b 5 cL implies a 63 c 5 b @ c and ( b @ c) 9 ( a 63 c ) = b 8 a. (iv) a 5 b 5 c implies ( c e b ) @ aE P , ( c e b ) @ ( b e a )E P and a@(c8b)= c e ( b e a )

(iii) a

and also ( c 8 b) @ (b 8 a ) = c €3 a. (v) a

2 b implies b = a @ (b €3 a).

(vi) a s b L a n d a s c L i m p l y a @ b = a @ c i f a n d o n l y i f b = c .

(vii) a s b 5 c L a n d c s d s a L i m p l y a @ d = b @ c i f a n d o n l y i f b 8 a = d @ c . (viii) a

5 bL 2 cL

implies a @ (b €3 c) = ( a @ b) 8 c.

(ix) c s a s b L a n d c s b imply ( a ~ c ) @ ( b ~ c ) = ( a @ b ) @ c e c .

Proof. We need to verify the axioms (Dl) - (D2) and the other properties of 8, which is routine. (i) a 5 bL is equivalent to b 5 aL, which implies aL 8 b 2 aL, therefore, a 2 (aL 8 b ) l = (a 63 b) and, moreover,

(ii) From the inequalities a Then

5 bL

and (a 63 b)

By symmetry, c 8 (a 63 b) = (c 8 a) 8 b. (iii) By the assumptions we have cL 8 b

5 c it follows that cL 2 (bL

8 a).

5 cL 8 a, hence,

and

(iv) I f a s b s c , t h e n c 8 b s c 8 a < l p 8 a = a L , therefore, a e ( c 8 b ) E P . By (ii) we have c 8 a 63 (c 8 b) = (c 8 a) 8 (c 8 b) = b 8 a, therefore,


125 l($,cp)12 and, since 11 p p oP$($)[I2= I ( $ I ( P > I ~ ~ 11 p p oP$]I2= I ( Y ! ' , ( P=>([(PI, ~~ Two states are said to be orthogonal or incoherent if the scalar product of their representative rays is zero. Actually, rays correspond to the secalled pure states of S, and general states are associated to positive trace class operators of class one, also called in the literature "density operators", "statistical operators", "density matrices". This set of operators is a convex subset of the vector space of all trace class operators: if S1and S2represent states then for any X E [0, 11,AS1 (1- X I S 2

+

Georges Chevalier

is also the representation of a state. As for pure states, we often identlfy a general state with its operator representation. Any pure state, considered as a rank-one projection, is an extremal point of the convex set of all states and any state is a linear combination of pure states: if S is a state then there exist a sequence of complex numbers (wi) and a sequence (w) of unit vectors such that wi = 1and S = wiPVi.

x

x

i

i

a An observable of S is a property that, in principle, can be measured. To

each observable corresponds a not necessarily bounded self-adjoint operator defined on H. a In the measurement of an observable represented by a self-adjoint operator

A, the spectrum a(A) of A defined by a(A)

=

{A E C I A - X . l H not invertible)

plays a crucial role. In particular, the possible values of a measurement of the physical quantity represented by A are all elements of a(A). Consider the simplest case of an observable represented by a self-adjoint operator A with a pure point spectrum. Let X be an eigenvalue of A and PA the orthogonal projection on the corresponding eigenspace. If, just prior to the measurement, the physical system S is in the pure state [cp], 11 cp )I= 1, then the outcome X is obtained with probability

If the outcome X is effectively obtained then the state after the measurement is [Px(cp)]. Note that if the measurement is immediately repeated, then the outcome X is attained with probability 1. For an observable with a spectrum containing a continuous part the situation is, roughly speaking, very similar but its rigorous mathematical treatment is more complicated and it uses the projection-valued form of the spectral theorem (see [~eltramettiand Cassinelli, 1981, Chapters 1 and 31 or [~vure~enskij, 1993, Chapter 11) Any state is also an observable and thus can be measured. In particular the possible measures of a pure state are 0 or 1. If the physical system is in the pure state [cp], then the probability of the outcome 1if an experiment is performed to determine if the system is in the pure state [$] is: Prob([$l = 1) =II Pdcp) /I2= ([$I, PI)^ (I1 cp II=II ?/1 II= 1). It is the square of the scalar product of the rays [cp] and [$I. If the outcome 1 is effectively attained then the state of the system immediately after the the transition measurement is [$I. This is the origin of the name of ( [cp] , probability between the pure states [cp] and [$I.

Wigner's Theorem and its Generalizations

433

Now a natural question is: Is any self-adjoint operator in correspondence with a physical quantity and does any ray represent a pure state of the physical system? In general the answer is no and the answer yes corresponds to strict quantum systems, i.e. systems without nonquantum (i.e. classical) features. In the presence of nonquantum features so called superselection rules must be introduced and the Hilbert space formulation becomes more complicated (see [~eltramettiand Cassinelli, 1981, Chapter 51). 3 THE ORIGIN OF WIGNER'S THEOREM AND A SHORT HISTORY OF ITS PROOFS The result known today as Wigner's theorem originally appeared in a book [Winer, 19311 written by E. Wigner in 1931. An English translation [Wigner, 19591 of this book, with three new chapters, was published in 1959. In the sequel, we briefly summarize the part of the book dealing with Wigner's theorem. we use the same notation and vocabulary as Wigner. In Chapter 6, entitled "Transformation Theory and the Bases for the Statistical Interpretation of Quantum Mechanics"', Wigner considers observables represented by operators G, G', G" ,. . .and wave functions cpl, cp2,. . . and he proves that the same results are obtained from this system of operators and wave functions as from the system in which the operators are replaced by

-

-

-

G = U G U - ~ ; G' = UG'U-~; GI' = ~

~"u-l I . . . ,

and the state functions by

-

-

pz = Ucpz; (P3 = Up3. . . , where U is an arbitrary unitary operator. First of all, the eigenvalues which define the possible results of the measurements of G and E = UGU-I are identical: if Xk is an eigenvalue of G, and qk the corresponding eigenfunction, then X k is also an eigenvalue of E and the corresponding eigenfunction is UQk. Moreover, the probability of the eigenvalue X k for the quantity corresponding to G in the first "coordinate system" and the one corresponding to ?? in the second are equal since cpl = Up1;

cp2

Similarly, the transition probabilities between pairs of corresponding states cpl, and Ucpl, Up2 are also the same in the two coordinate systems, since

Wigner calls such a transformation of the coordinate system by a similarity transformation of the operators and simultaneous replacement of the wave functions cp by U p a canonical transformation. Two descriptions which result from one another by a canonical transformation are equivalent. The converse of this is the topic of the Appendix of Chapter 20, which is devoted to the study of the equation

Georges Chevalier

where and 5 are the wave functions ascribed by the second observer of a physical system to the states which the first observer describes by Q and a. Wigner proves that, up to a phase factor, = U(cP) with U being a unitary or antiunitary operator. By considering two stationary states with diierent energies he shows that the operator U is actually unitary. Wigner notes that for the exclusion of the antiunitary case he used the time dependence of the states. More precisely, he postulated that if state is transformed into state cP' in the course of the time interval t, then state is transformed into during the same time interval. In Chapter 26, "Time Inversion", added in the English translation of his book [Wigner, 19591, Wigner considers a physical system with zero linear momentum and the transformation t -t -t which transforms a state cp into a state 8(p) in which all velocities have opposite directions to those of cp. He proves that 8 satisfies Equation (2) and corresponds to an antiunitary operator. The chapter continues with a general study of time inversion and antiunitary operators. A good reference concerning time inversion, also called time reversal, and antiunitary operators is [ ~ o u t a ~et~ al., e l 1965, expecially Section 4.101. See also [Henley,20011, [Uhlhorn, 1962, Section 91. Wigner's original proof is not complete. Nevertheless it represents "an excellent basis for an elementary and straightforward proof' [Bargmann, 19641. In his 1962 paper [Uhlhorn, 19621, U. Uhlhorn gave a proof using new ideas and, moreover, discussed the proofs that had been given until 1962 by E. Wigner, R. Hagedorn, J . M. Jauch and G. Ludwig. All these proofs are incomplete or incorrect and thus it seems that hi paper contains the first proof of Wigner's theorem. This proof is analysed in Section 5. After Uhlhorn's proof numerous proofs of Wigner's theorem appeared in the literature. We can mention the following authors: J . S. Lomont and P. Mendelson [1963]: an elementary proof apart the use of the following result of analysis: a bounded subset of a Hilbert space is weakly sequentially compact. Thii result can be obtained as a consequence of the reflexivity of Hilbert spaces.

G. Emch and C. Piron 119631: a proof in the framework of lattice theory and projective geometry. See also iron, 19761, Theorems 3.23 and 3.24. This proof is summarized in Section 6.

V. Bargman 119641: an elementary proof in the spirit of the original work of Wigner. Contrary to Uhlhorn, Bargman uses the same proof if dim H = 2 and if dim H >_ 3. For this reason, his proof is often considered as the first correct proof of Wigner's theorem.

Wigner's Theorem and its Generalizations

435

S. C. Sharma and D. F. Almeida [1990b]. Their proof forms the subject of our Section 4. M. Gyory [2004]. This is the last known proof in 2005. It uses Zorn's lemma. See also: [Simon, 1976; Varadarajan, 1985; Zhmud, 1992; Weinberg, 1995; Ratz, 1996; Cassinelli et al., 19971. Of course, proofs of generalized versions of Wigner's theorem can also be considered prooh of the original theorem [Wright, 1977; Molnir, 1996; Molnir, 1998; Molnk, 1999; BakiC and Guljas, 2002; Chevalier, 2005bl in chronological order). 4

AN ELEMENTARY PROOF OF WIGNER'S THEOREM

The purpose of this section is to give a simple proof of Wigner's theorem in full detail and, as V. Bargmann noted in [1964], to emphasize its "quite elementary nature". Our proof is close to the proof given in [ ~ h a r m aand Almeida, 1990bl. This latter proof is itself similar to Bargmann's [~argmann,19641 which is, two years after the proof of Uhlhorn, one of the first complete and rigorous proofs of the theorem. Before giving a precise statement of Wigner's theorem we need some definitions. Let H be an inner product space over K = B or C. 1. A mapping U : H + H is an isometry if (U(x), U(y)) = (x, y) for all x,

y E H . Any isometry is an injective operator. If U is onto and H a Hilbert space then U is invertible with U-l = U*, the adjoint of U. In this case, U is called a unitary operator. 2. If K = C, an additive mapping U : H + H is said to be antilinear or conjugate linear if U(Xx) = XU(Z) , x E H, X E C. If, for any x, y E H, (U(x), U(y)) = (y, x) then U is an anti-isometry. In a Hilbert space, a surjective anti-isometry is called an antiunitary operator. Unitary and antiunitary operators satisfy U-I = U*. More generally, let El and E2be two vector spaces over the fields K1 and K2 and a : Kl + K 2 be a field isomorphism. An additive mapping f : El + E2 is said to be semilinear or a-linear if, for any X E Kl, any x E El,

3. Two mappings T : H -t H and TI : H -t H differ by a phase factor or are phase factor equivalent if there exists cp : H + K such that for any x E H, T(x) = cp(x)T1(x) with Icp(x)I = 1. Remark that the binary relation "to be phase factor equivalent" is an equivalence relation.

Georges Chevalier

436

4. Let f be a mapping from a set X into a set Y and let D be a subset of P(X), the power set of X. A mapping F : D + P(Y) is said to be induced or generated by f if for every M E D l F ( M ) = { f (m) I m E M}. As usual, this last set is also noted f (M) . 5. A bijection T : H 4 H is called a symmetry if T preserves the modulus of the inner product:

Every symmetry T generates a bijection f' of [HI,defined by f'([cp]) = [T(cp)], cp E H, which preserves the scalar product of rays. A proof of this claim uses the forthcoming Lemma 1. A bijection of [HI preserving the scalar product of rays is also called a symmetry in the literature. Let ? be such a symmetry and choose in every ray X a unit vector u(X). Define T : H + H by T(0) = 0 and if x # 0 then x = Xu(X) for a unique ray X and T(x) = Xu(T(X)). The mapping T is a bijection of H preserving the modulus of the scalar product and generating the same bijection of [HI than ?. When it is useful to avoid ambiguity, this second kind of symmetry will be called, as in [~aradarajan,19851, a physical symmetry. (T(x),T(y)) = 0 for a bijection T of H then T is called an If (x, y) = 0 I-symmetry (ortho-symmetry). As for symmetries, any I-symmetry generates a bijection of [HI preserving the orthogonality of rays in both directions (use the forthcomming Lemma 7 for a proof) and, conversely, a bijection of [HI preserving orthogonality in both directions allows to define a I-symmetry. In the literature, a bijection of [HI preserving the orthogonality of rays is also called an I-symmetry. Recall that a mapping which differs by a phase factor from a symmetry or an I-symmetry is also a symmetry or a I-symmetry respectively. A first lemma [Sharma and Almeida, 1990b, Lemma 51 (see also [Wigner, 1959, page 2341) will show that a mapping defined on an inner product space and preserving the modulus of the inner product has a property close to linearity. LEMMA 1. Let H be a complex inner product space and consider a mapping f from a subset D of H into H which preserves the modulus of the inner product. For any pair x and y of mutually nonzero orthogonal vectors of D and any pair of complex numbers a and p such that a x py E D there exist a', p' E C such that

+

f (ax Moreover, Ial = la'l and

+ Py) = ff'f(XI+ P'f (Y),

Ipl = IP'I.

Proof. Assume P = 0 and a # 0. The Schwarz equality states that two vectors u and v in an inner product space are linearly dependent if and only if 11 u 11 11 u I[= 1 (u, v )1. Thus a mapping preserving the modulus of the inner product

437

Wigner's Theorem and its Generalizations

also preserves the linear dependence of two vectors. Since x and a x are linearly dependent, there exits a' E C such that f ( a x ) = a'f ( x ) . We have

and thus la1 = ldl. Now if x and y are orthogonal unit vectors then f ( x ) and f ( y ) are also orthogonal unit vectors and, by the Fourier expansion, the claim of the lemma is true if and only if

By a straightforward calculation ( f ( a x PY) - ( f ( a x PY), f ( x ) ) f($1 ( f ( a x + f l y ) ,f ( Y ) ) f ( Y ) , f ( a x PY) - ( f ( a x PY), f ( x ) ) f(XI + ( f ( a x + PY),f ( Y ) ) ~ ( Y ) = ) 0 and thus Equation (3) is satisfied. We have

+

+

+

+

+

l(f(ax+Py),f(x))l = I(ffx+P~,x)I=lalIIxIl~ = I(alf(4 P'f ( Y ) , f ().I = la'l

+

II f (4112=

la'l

and thus Ial = la'1 . Similarly, IPI = IP'I . The general case is obtained by combining these two particular cases.

II 112 rn

REMARKS. 1. If f is defined on the whole of H, then the forthcoming Lemma 7 allows one to obtain a shorter proof. 2. Iff preserves the modulus of the inner product then, by the previous lemma, the image of the line spanned by a nonzero vector x is contained in the line spanned by f ( x ) and the image of the plane spanned by two mutually orthogonal vectors x and y is contained in the plane generated by f ( x ) and f (y) but, in general, the image by f of a line or a plane is not a line or a plane respectively. For example, choose a unit vector u ( D ) on any line D and let f : H + H be the mapping defined as follows:

f (0)= 0, if x f 0 then let D be the line generated by x and if x = Xu(D) set f ( X I =I l u(D). Clearly, f preserves the modulus of the inner product and the image by f of the line Cu(D)is B+u(D)which is not a line. Recall that f is not linear or antilinear since f ( x ) = f ( i x ) and thus a mapping preserving the modulus of the inner product is not necessarily linear or antilinear. Wigner's theorem can now be stated as follows.

Georges Chevalier

438

PROPOSITION 2. Let H be a complex inner product space with dim H 2 2 and T : H 4 H a mapping presenring the modulus of the inner product. There exists an isometry or a n anti-isometry A on H which diflers from T only by a phase factor. T w o isometries A and A' or two anti-isometrics B and B' satisfy the requirement of the preceding sentence if and only if they difler by a constant phase factor. The mapping T cannot be phase equivalent to an isometry and to a n anti-isometry. If T is a symmetry (i.e. if T i s surjective) and H a Hilbert space then A is a unitary or an antiunitary operator.

Proof. The proof splits into six steps. Step 1: Definition of a mapping A which differs from T by a phase factor. Let xo be a unit vector in H chosen once and for all, X = @so the line spanned by xo and let D = (xo X I ) U X I . Note that ( x o X I ) n X I = 0. First we will try to define a mapping A : D + H which differs from T by a phase factor and such that:

+

+

If A exists then A preserves the modulus of the inner product. Thus for any y E X I and any a E @, A ( a y ) = a t A ( y ) with la]= la'[. For any mapping f : D -+ H which preserves the modulus of the inner product and any y E X I , ( f ( s o Y ) - f ( s o ) - f ( Y ) ,f ( s o Y ) - f ( s o ) - f ( Y ) ) = 1+ II Y 112 +1+ II Y 112 - ( f ( 2 0 + Y ) , f(xo>>- ( f (xo)1f (xo + Y ) ) - ( f (xo + Y ) , f ( Y ) )( f ( y ) ,f ( s o y ) ) . Therefore, for the satisfaction of ( 4 ) by f , it suffices that ( f (xo y ) , f (so)) = 1 and ( f (so+ y ) , f ( y ) ) =I] y 112. We will prove that such a mapping exists.

+

+

+

+

If y E xo

+ X ' - define

We have A ( x o ) = T ( x o ) , ( A ( y ) , A ( x o ) )= 1 and

I(T(Y),T(xo)>l= I(Y,xo)I If y E X I then xo

=

I(x0

+

1

I

)I

T(xo) (Y - xo),xo)I = 1(xo,xo)I = 1.

= 1 since

+ y E xo + X I and, by the previous definition,

and so (A(xo+y ) , A ( x o ) ) = 1. Thus for the satisfaction of (4) by A it suffices to find X E @ such that 1X1 = 1 , to define A ( y ) = A T ( y ) and to check that ( 4 x 0 Y ) ,A(Y))=11 Y 112. - ( T ( x 0 Y ) ,T ( Y ) ) If A ( y ) = A T ( y ) we have ( A ( x 0 y), A ( y ) ) = X (Tho +I, T ( x 0 ) )'

+

+

+

+

Wigner's Theorem and its Generalizations

Thus X

=[I

y

(I2

(T(~0)lT(xo + y)) is convenient since (T(Y),T(xo + Y))

We have succeeded in defining A : D + H which differs from T only by a phase factor and which fulfills (4). Step 2: The restriction of A to a line of

XIis

linear or antilinear.

Let y E x'- a unit vector. By Lemma 1,for any a E C there exists a' E C such that A(ay) = a1A(y) with la1 = la'l. Using Equation (4), we have

On the other hand, I(A(x0

+ ffy),A(xo+y))l = I(xo + a y , x o +Y)I = I1 + al

+

and so 11+ a1 = 11 all. Therefore 1+ a + E a E = 1+ a' + 2 0'2and, since la1 = la'l, a E = a' + 2. Thus a and a' have the same real part. Since they have also the same modulus, their imaginary parts - are equal or opposite and finally, a = a' (Referred as Condition (PI)) or a = a' (Referred as Condition (Pz)). Note that the end of the previous proof shows that:

+

+

+

LEMMA 3. Let a and a' be two complex - numbers with the'same modulus. If Il+al = Il+alI then a = a' or a = a'. (This Lemma is used by Wigner in his proof, see [Wigner, 1959, page 2341.) Now we will prove that for any unit vector y E XI,Condition (PI) holds for any a E C or Condition (P2) holds for any a E C. Let a, /? E C*. By Lemma 1, there exists a', P1 E cC such that la1 = la'[, IPI = IP'I and A(ay) = a1A(y), A(Py) = PtA(y). Using Equation (4), A(xo+ay) = A(x0) A(ay) = A(x0) a1A(y) and, similarly, A(x0 Py) = A(x0) PIA(y). Thus ,

+

+

+

I(x0 + PY,XO

+ ~ Y )=I I(A(xo),A(xo)>+ (PIA(y),a1A(y))I I1 +Wl=

ImI

+

I1+P1a'l

= But = [@'TI and, using Lemma 3, or ,BE = Fa'. Assume that a satisfies (PI) and p satisfies (P2) then = @ or @ = Pa. In the first case, ,8= and @ also satisfies ( 4 ) . In the second case a = E and a also satisfies (P2). In both cases, a and P satisfy the same condition.

p

m

440

Georges Chevalier

Finally, the restriction of A to a line of X I is linear or antilinear (In the definition of a linear mapping f on a Hilbert space, it suffices to assume that f (Ax) = Af (x) for any unit vector x).

Step 3: The restriction of A to X I is additive.

A proof is necessary only if dim H > 2. Let y and z be two orthogonal unit vectors in X I . For all a and p in C, a y + p z E XI.Using Equation (4) and Lemma 1, A(xo + (ay + pz)) = A(xo) + A(ay + pz) = A(xo) + al'A(y) + pl'A(z), Ial = Ia1'I and IpI = IP1'I. Let a' and $ be such that A(ay) = alA(y) and A(@z) = PIA(z) with la1 = \all and = I/?'I.

Since IaEl = 1aU21,we can use Lemma 3 and we get aE = aN2.Since a"7 E W+, a' and a" we get by the same argument that Ia'l = la"/implies a' = a". By similar reasoning we get p1 = P" and

Let u and v be two nonzero vectors in x'- and let w be a unit vector orthogonal t o u in the subspace spanned by u and u. There exist a and ,B in C such that

In the following calculation, if the restriction of A to Cu is linear then & = cr and if this restriction is antiliiear then & = E

We conclude that the restriction of A to X I is additive. Step 4: The restriction of A to XIis linear or antilinear and A is extended to H.

Wigner's Theorem and its Generalizations

44 1

This result is already proved if dim H = 2. Thus we assume dim H > 2. Let yl and y2 be two nonzero and noncollinear vectors in X I and assume that the restriction of A to (Cyl is linear while its restriction to Cy2 is antilinear. If yl is orthogonal to y2 then

which is absurd. Now if yl and y2 are not orthogonal then let z be a vector orthogonal to yl in the subspace spanned by yl and y2. There exist a and P E C such that yz = a y l pz. By the beginning of the proof, the restriction of A to Cz is linear and for all X E (C

+

In particular, A(iy2) = iA(y2) and also A(iy2) = -iA(yz) since the restriction of A to Cy2 is antilinear. Therefore A(y2) = 0, which contradicts the hypothesis of preservation of the modulus of the inner product. The restriction of A to X I is linear or antilinear. If this restriction is linear, we extend it to H by linearity and, if this restriction is antilinear, the extension is by antilinearity. Now, A is defined in the whole of H and, since A(x0) = T(xo), is phase equivalent to T. Step 5: A and T differ by a phase factor, A is an isometry or an anti-isometry and A is uniaue UD to a constant ~ h a s efactor.

Let ax0 + p y E H = x@x-'-. Since A(py) differs from T(py) by a phase factor, we can assume a # 0 and write in the linear case:

In the antilinear case, E-' replaces a-I and we have 171 = 1 and 101 = lall. Therefore laI7a-l = Id7~-l= 1and thus T(axo py) and A(axo +fly) differs by a phase factor. Note that, since T is norm preserving, A is norm preserving too and, since A is linear or antilinear, A is bounded and therefore continuous.

1

+

1

Let y, z E H. We have (A(Y - z), A(Y - 2))

=II

A(Y) 112

+ II A(%) 112

- M y ) , A(%))- (A(%),A(Y)

and also (A(y - ~ ) , A ( Y - 2)) = (Y- 2, Y - 4 and therefore,

=11

Y

112

+ II z 112

-(y, 2) - (2, y)

Georges Chevalier

442

(5)

(Y,z)

+ (z, Y) = (A(Y),

+ (A(%),A(Y))

Replacing z by iz in (5) allows one to obtain (y, z) case and (y, z) = (A(z), A(y)) in the antilinear one.

=

(A(y), A(z)) in the linear

In order to make precise the uniqueness of A, we separate out a lemma which is of interst in its own right. LEMMA 4. Let H be a complex inner product space with dim H 2 2 and T : H + H a mapping preserving the modulus of the inner product.

I. Let A be an isometry or an anti-isometry phase equivalent to T. Then the nature of A can be predicted fi-om T.

2. An isometry cannot be phase equivalent to an anti-isometry. 3. If two isometries or two anti-isometrics are phase equivalent then they difler by a constant factor. Proof. As in [~argmann,19641, consider three rays [pl], [pa]and [p3]generated by unit vectors p l , p2 and p3. The expression (pl, p2) (p2, p3) (p3,pl) is independent of the choice of the unit vectors generating the rays and is a function A([cpl], [pz], [p3]) of the rays [pi].Note that

if T is phase equivalent to TI then

Therefore, if there exist three unit vectors el, e2 and e3 such that A([el], [ea],[es]) is not real, an isometry cannot be phase equivalent to an anti-isometry and T is phase equivalent to an isometry A if and only if A(T([el]),T([ez]),T([e3])) = A([el], [ez],[e31> Since dimH 2 2, let e and f be two orthogonal unit vectors and consider the 1

1

unit vectors el = e, e2 = -(e

a: $

- f ) and e3 = -(e

6

+ (1 - i)f) [Bargmann, 19641.

We have A([el], [el], [e3]) = and the proof of 1. and 2. is complete. (For another approach to the distinction between unitary and antiunitary operators, see [Wigner, 19601.) If A and B are two isometries or two anti-isometrics phase equivalent then, for every line p in H, A(p) = B ( p ) . A classical exercise in elementary linear algebra states that if f and g are two linear mappings defined on a vector space E over a field K and if, for every line p, f (p) = g(p) then there exists X E K such that

Wigner's Theorem and its Generalizations

443

f = Xg. The result is true as well for antilinear mappings and if E is a Hilbert space and f and g are norm preserving then IXI = 1. Therefore two isometries or two anti-isometries which are phase equivalent to T d 8 e r by a constant of modulus 1.

To sum up, we have defined an isometry or an anti-isometry A which is phase factor equivalent to T and the result is valid for any inner product space of dimension greater than 2. S t e p 6: If T is a symmetry and H a Hilbert space then A is a unitary or an antiunitary operator. Let 0 # z E H. If T is surjective, there exists x E H such that 1 T ( x ) = z and A(%) = XT(x) with X E C*. Thus A(-x) = z if A is linear and X 1 A(:%) = z in the antilinear case. The mapping A is surjective.

X

If H is a Hilbert space, then the adjoint A* of A is a mapping from H to H which is linear or antilinear if A is linear or antilinear respectively. Let y, z E H. If A is linear

and, in the antilinear case,

and thus, in the two cases, A*A = l H . (Information about adjoints of semilinear mappings can be found in Section 10) Since A is bijective, A-' exists and

If the mapping A is linear then A is a unitary operator and, otherwise, A is an antiunitary operator. The following corollary is useful for the understanding of generalizations of Wigner's Theorem or its formulation in the framework of projections (for example, see [Wright, 19771).

COROLLARY 5. Let T be a bijection of the set of all rank-one projections of a Hilbert space H . If t r ( P Q ) = t r ( T ( P ) T ( Q ) )then there exists a unitary or antiunitary operator U : H + H such that T ( P )= UPU*,

444

Georges Chevalier

We begin the proof by a remark: if U is a unitary or antiunitary operator then, ) for any closed subspace M of H, P U ( ~=) UPMU*where PM and P U ( ~denote the orthogonal projections on M and U ( M ) . Indeed,

and clearly UPMU*is an orthogonal projection. Using Equation (1)in Section 2 we see that the bijection T induces a bijection of [HI preserving the scalar product of rays. The later bijection generates a bijection of H preserving the modulus of the inner product. Let us denote these three bijections by the same symbol T . By Wigner's theorem, there exists a unitary or antiunitary operator U : H + H such that, for any unit vector cp E H, T(cp) = X(cp)U(cp)with IX(cp) 1 = 1. We have [T(cp)]= [U(cp)] and thus, using the hypothesis concerning the symbol T ,

COROLLARY 6. The set of all physical symmetries of H forms a group isomorphic t o the quotient group of the group of all unitary or antiunitary operators by = 1). i t s subgroup {XIH I The set of all physical symmetries is clearly a subgroup of the group of all bijections of [HI. Let f be the mapping £rom the group of all unitary or antiunitary operators into this subgroup which associates to each U the physical symmetry induced by U . The mapping f is a group homomorphism and Wigner's theorem means that f is onto. Its kernel is the set of all unitary or antiunitary operators U satisfying, for any line p of H , U(p) = p. By using Lemma 4, kerf = {XIH I 1x1 = 1). The group of all physical symmetries of H is called the symmetry group of H or of the physical system associated to H . The symmetries, considered a s mapping from H to H, form a group as well. By Wigner's theorem the binary relation "to be phase equivalent" is a congruence relation and the quotient group is isomorphic to the symmetry group of H. Some other groups isomorphic to the symmetry group will be studied in Section 11 REMARKS. 1) Let H be an inner product space over K = W or C and f : H + H . Consider the two properties:

Wigner's Theorem and its Generalizations

445

+

I f f satisfies 1. or 2. then a straightforward calculation yields to ( f ( x Y) f (x) - f ( Y ) ,f ( x + y ) - f ( x ) - f ( y ) ) = 0 and therefore f is additive. If f satisfies 1. then, for any X E K, ( f ( X x ) - X f ( x ) ,f ( X x ) - X f ( x ) ) =Oand f is an isometry and, similarly, iff satisfies 2. and K = C, f is an anti-isometry. Moreover, if f is onto and H a Hilbert space, then f is a unitary operator when f fulfills 1. and an antiunitary operator if K = C and f satisfies 2..

2) A linear mapping which preserves the inner product is not necessarily onto in the infinitedimensional case. For example, let (en)nENbe an orthonormal basis of a separable Hilbert space H. The shift operator x = Enen 3 nEM

is a non surjective operator preserving the inner product.

S ( x )= nEN

In the finite dimensional case, any operator preserving the inner product is a bijection since it is a one-bone operator and the dimension of the vector space is finite. 3) In [Sharma and Almeida, 1990b]the authors give a simple proof that an additive mapping preserving the modulus of the inner product is linear or antilinear. They use a result of [pian and Sharma, 19831: An additive bounded mapping f is a sum of linear operator and an antilinear operator. By using the first claim, the step 4 in the proof of Wigner's theorem is unnecessary.

4 ) ) Let T be a symmetry on a one-dimensional Hilbert space and let xo be a unit vector. Define two mappings A : H + H and B : H + H by A(Xx0) = XT(xo) and B(Xx0) = ~ T ( X ~for) any X E C. It is easy to check that A is an isometry, B is an anti-isometry and that T, A and B differs by a phase factor. A part of Lemma 4 is no longer true if dimH = 1. 5) The original proof of W i n e r [1959, Appendix of Ch. 201 and the proof of Bargmann [1964]use a Hilbertian basis and thus cannot be done in an inner product space. 5 UHLHORN'S VERSION OF WIGNER'S THEOREM The main idea and the great significance of the paper [Uhlhorn, 19621 is well summarized by the following part of its Introduction:

W e shall replace the requirement of the invariance of transition probabilities by the requirement that orthogonal vector rays are transformed into orthogonal vector rays, that is, incoherent states are transformed into incoherent states. B y this definition, a symmetry transformation i s a mapping preserving the logical structure of quantum mechanics, whereas the definition stated above corresponds to a mapping preserving the probabilistic structure of quantum mechanics.

Georges Chevalier

446

In his proof of Wigner's theorem, Uhlhorn considers bijections of lines preserving linear independence and a lemma is useful to understand the connection between this notion and the notion of bijections of lines preserving orthogonality. By definition, n lines in a vector space E, 11, . . . ,In, are said to be linearly independent n

li is a i=l direct sum of subspaces. This also means that 11,. . . ,ln considered as elements of the modular lattice of all subspaces of E are independent [Birkhoff, 1967, Ch. IV, 941 LEMMA 7. if they are generated by linearly independent vectors or equivalently if

+

1. I n a n inner product space H, n 1 vectors XI, . . .,xn+l are linearly independent exactly if the n first vectors are linearly independent and there exists a vector y which is orthogonal to the n first vectors and non-orthogonal to xn+ 12. Every mapping j : H 4 H preserving orthogonality of vectors in both directions also preserves linear independence of vectors in both directions and its extension to the set of all lines of H preserves independence of lines in both directions. Proof. 1. Assume that X I , . . . ,xn+l are linearly independent. Let X I , y2,. . . ,yn+l be the n 1 vectors obtained from X I , . . . ,xn+l by using the Gram-Schmidt orthonormalization process. It is easy to check that y = yn+l is convenient. Con-

+

versely, assume that the n first vectors are linearly independent and there exists a vector y which is orthogonal to the n first vectors and non-orthogonal to %,+I. If Xlxl . . . Xn+lxn+l = 0 then ( y , Xlxl . . . Xn+lxn+l) = 0 implies An+l = 0. Since the n first vectors are linearly independent, X1 = . . . = An = 0 and X I , . . . ,xn+l are linearly independent. The claim 2. is an easy consequence of 1.

+ +

+ +

In Euclidean spaces and, more generally, in inner product spaces, the preservation of linear independence of lines is an hypothesis strictly weaker than the preservation of orthogonality of lines. For example, let j be an isometry of the Euclidean space Rn equipped with its canonical scalar product. If one changes the scalar product then f always preserves linear independence of lines but not, in general, orthogonality of lines with respect to the new product. We can now state the main result of [Uhlhorn, 19621 as follows. PROPOSITION 8 ([Uhlhorn, 1962, Lemma 3.4, Theorems 4.1 and 4.21). Let H and H' be two complex Hilbert spaces with d i m H 2 3. If T : [HI + [H'] i s a n I-symmetry then there exists a unitary or antiunitary operator O : H -+ H' which induces T. The operator O is determined by T up to a complex factor of modulus 1.

Wigner's Theorem and its Generalizations

447

Shortened proof. Using the Axiom of Choice, there exists r : C + C such that T(C x) = C r(x) if x # 0 and r(0) = 0. Let x and y be two linearly independent vectors. Since x, y and x y are linearly dependent, the same holds for r(x), r(y) and r ( x y). There exists a mapping w defined for any pairs of linearly independent vectors and with values in C such that

+

+

x) instead of w(y, x) is useful for the simplicity of the forthcoming (To write W(X+Y, Equation (7)). Using (6) and r((x y) z) = r ( x (y z)), we have for arbitrary linearly independent vectors x, y and z (Remark we use dim H 3)

+ +

+ +

>

Another relation for linearly independent vectors x, y is w(x, y)w(y, x) = 1. The definition of w extends to pairs of linearly dependent vectors in such a way that (7) becomes valid. This part of the proof is based on the following fact: if z is linearly independent of the linearly dependent vectors x, y then w(x, z)w(z, y) is independent of z and allows to define w(x, y). Now choose 0 # xo E H and define 8 : H 4 HI by 8(x) = w(xo,x)r(x) if x # 0 and 8(0) = 0. This mapping is bijective, additive and, since C 8(x) = C r(x) = T(C x), 8 induces T. For 0 # a E C and 0 # x E H , C x = C(ax) and therefore 8(x) and 8(ax) generate the same ray in HI. So we can consider cp : C* x ( H - (0)) + C defined by 8(ax) = cp(a, X ) ~ ( X )For . any fixed 0 # x E H , a + cp(a, x) is a mapping cp, : C + C. Using

it is easy to see that all the mappings cp, coincide. Let +(a)= cp(a,x) for an arbitrary nonzero x H with +(O) = 0. The mapping 4 is a nonzero homomorphiim of the complex field. Let x and y be two orthogonal vectors in H with the same norm. For a # 0, x - ~ - l y and x + a y are orthogonal vectors and therefore 8(z-E-ly) and B(x+ay) are also orthogonal (It is the first time we use the preservation of orthogonality of lines. So far, we have only used the preservation of linear independence of lines). This yields to

11 8(x) ]I2 - + ( a ) + ( ~ - l ) 11 8(y) 112=

-

0 and +(a)+(-E-l) =

11 e(x>[I2 II 8(Y) 112

+

is independent of a. By choosing a = 1, we have +(E) = +(a). Thus is the identity or the conjugation and 8 is linear or conjugate linear. Let 6 : ( H - (0)) + cC be the mapping defined by 11 8(x) I]= 6(x) 11 x 11 if x # 0. If x and y are two non-orthogonal vectors then z = (y, y)x - (x, y)y is orthogonal to y and thus we have

448

Georges Chevalier

By interchanging x and y and taking the complex conjugate

and combining (8) and (9) yields b ( x ) = 6 ( y ) . The same is true if s and y are orthogonal. Let 6 = 6 ( x ) for a non-zero x. Using equation ( 8 ) we have 1

( e ( x )8(y)) , = h 2 4 ( ( x ,y ) ) . Finally Q = 1 0 is an isometry or an anti-isometry. If T 6 is onto and H a Hilbert space then Q is a unitary or anti-unitary operator which induces T. REMARKS. 1) A careful study of the proof of Theorem 4.1 in [Uhlhorn, 19621 shows, as Uhlhorn noted himself, that it gives rise to a proof of the following version of the First Fundamental Theorem of projective geometry. PROPOSITION 9. Let El and E2 be two vector spaces over the commutative fields K1 and K2 with dim El 2 3. If there exists a bijection T from the set of all the lines of El onto the set of all the lines of Ez which preserves linear independence of lines then in both directions: 1. the two fields K 1 and K2 are isomorphic;

2. there exists a semilinear bijection s : El -+ E2 inducing T In [ ~ h l h o r n19621 , the surjectivity of the homomorphism 4 of C is not proved since the last part of the proof of Proposition 8 implies that 4 is onto if T preserves orthogonality of lines. With the weaker hypothesis saying that T preserves linear independence of lines the proof of its surjectivity is not difficult and it is similar to a part Baer's proof of the First Fundamental Theorem of projective geometry [ ~ a e r1952, , page 501. from the following Lemma we see that Proposition 9 is actually not an improvement of the First Fundamental Theorem of projective geometry. LEMMA 10. A bijection cp of the set of all lines of a vector space E over the field K preserves linearly independence of lines in both directions if a n only if cp extends t o a n isomorphism @ of the lattice of all subspaces of E. Proof. An isomorphism of the lattice of all subspaces of E preserves, in both directions, the dimension of subspaces and therefore the linear independence of lines. Now let cp be a bijection of the set of all lines of E which preserves linear indepencp(1). If dence in both directions. Define for any subspace F of E , @ ( F )=

V

1 line,1CF

F c G then @ ( F )c @ ( G )and conversely assume that @ ( F )c @ ( G )for two subp(1) = spaces F and G . Let m C F be a line. We have p ( m ) C @ ( G )=

V

1 line,1CG

Wigner's Theorem and its Generalizations

449

~ ( 1 ) If . p(m) = K a, a E E, there exists n linearly independent vectors, 1 line, 1CG

a l , . . . ,a, such that a = a1 + . . . + an with ai E cp(li), li a line of G. We have cp(m) c cp(ll) . . . cp(ln) and, as {cp(ll), . . . ,cp(1,)) is a linearly independent set of lines and {cp(m), cp(ll), . . . ,cp(1,)) is not linearly independent, (11,. . . ,I,) is linearly independent and {m, 11, . . . , 1,) is not linearly independent. Therefore m C l1 @ . . . @ 1, and m is a line of G. Thus F c G and @ is an isomorphism of the lattice of all the subspaces of E. 2) Uhlhorn's version of Wigner's theorem is invalid for twedimensional Hilbert spaces. A first example of a ray transformation defined on a two-dimensional Hilbert space which preserves orthogonality of lines but not all transition probabilities is given by Uhlhorn in [1962] and a simplified version of this example may be found in [Cassinelli et al., 19971. In [1962, Theorem 5.11, Uhlhorn states Wigner's theorem in the two dimensional case for a ray transformation preserving transition probabilities. He has thus obtained the first complete proof of the usual version of Wigner's theorem. 6

WIGNER'S THEOREM VIEWED BY THE GENEVA SCHOOL

In the early 1960s, in Geneva, J. M. Jauch and especially C. Piron developed, jointly with some other mathematicians and physicists, a new formulation of quantum mechanics [Jauch, 1968; Piron, 19761. Some other mathematicians (D. Aerts, I. Daubechie, B. Coecke,. . .) continued this work in Brussels. This approach to the foundations of quantum mechanics is known as the Geneva School or the Geneva-Brussels School. We will outline only that part of this. formulation which is relevant to the understanding of Wigner's theorem. Every measurement on a physical system can be reduced, at least in principle, to the measurements of a series of yes-no experiments which, in a sense, represent propositions. These experiment are observations permitting only one of two alternatives as an answer [Jauch, 1968; Piron, 19761. The set of all such propositions of a physical system is a complete atomic orthomodular lattice satisfying the exchange axiom p iron, 1976, Chapter 21 and [Kalmbach, 19831 or [ ~ a e d aand Maeda, 19701 for information about this kind of lattices). It is called a propositional system. Any propositional system is isomorphic to a product of irreducible propositional systems. The following result gives a representation theorem for such structures. It also motivates the use of Hilbert spaces in the formalization of quantum mechanics. We need a definition. Let E be a vector space over a field K equipped with an involutorial antiautomorphism X + A* and let (., .) be a mapping from E x E into K. If, for x, y, ZEE,AEK,

Georges Chevalier

3. (x,x) = 0 implies x

= 0.

then (., .) is called a definite Hermitian form. A subspace F of E is said to be closed or I-closed if F = FLL. PROPOSITION 11 p iron, 1976, Theorems 3.23 and 3.241). I . Let L be a n irreducible propositional system of height 2 4. The orthocomplemented lattice L is isomorphic to the lattice of all closed subspaces of a vector space E equipped with a definite Hermitian form.

2. Conversely, the lattice of all closed subspaces of a vector space E equipped with a definite Hermitian f o m is a n irreducible propositional system if and only if, for any closed subspace F,

A vector space E equipped with a definite Hermitian form satisfying Equation (10) is said to be a generalized Hilbert space. Any classical Hilbert space over R, C or IHI is a generalized Hilbert space but there exist non classical Hilbert spaces [Keller, 19801. In iron, 19761, the following result is called Wigner's theorem. PROPOSITION 12 s iron, 1976, Theorem 3.281; see also [ ~ m c and h Piron, 19631 and [Jauch, 1968, 9.41). Let (El, (., .)l) and (E2, (., .)2) be two generalized Hilbert spaces of dimensions at least equal to 3 and let Li, i = 1, 2, be the irreducible propositional systems of all closed subspaces of Ei. A n y isomorphism Q, of L1 onto L2 i s induced by a semilinear bijection of El onto E2. Conversely, a a-linear bijection f : El + E2 induces a n isomorphism of L1 onto L2 i f and only i f there exists a n element a in the scalar field K1 of El such that

Idea of the proof. The isomorphism Q, extends to the lattice of all subspaces of El by Q,(F) =

V @(PI

p line, pCF

for all subspaces F of El. By the First Fundamental Theorem of projective g e ometry [Baer, 19521 there exists a semilinear bijection f : El + E2 inducing Q,. Conversely, if a semilinear bijection satisfies (11) then

and thus, for any subspace F c El, f (FL) = f ( F ) ~ . The image of a closed subspace of El under f is a closed subspace of E2 by Equation (11) and Equation

Wigner's Theorem and its Generalizations

45 1

(11) is sufficient for the preservation of the structure L1 and L2 as orthocomplemented lattices. As the two forms (., and (x, y ) + a-'((f (x), f ( y ) ) ~define ) the same orthocomplementation on L1, a classical result of Birkhoff and von Neumann [I9361 or [Varadarajan, 1985, Theorem 2.61 shows the necessity of Equation (11). Let * and be the involutorial antiautomorphisms of K1 and K2 associated to and (., .)2. Replacing y by Xy in Equation (11) yields (., (12) a(X)* = a(a-lX*a). If Kl and K2 are both the field of complex numbers and * and agree with the usual conjugation then, by Equation (12), a(X) = a(1) and thus a(R) = R. The automorphim a is either the identity or the conjugation. Moreover x = y in Equation (11) implies that a is a positive real number. Define u =

-,f

6

then u

induces the same isomorphism than f and the following corollary is proved. COROLLARY 13 ([Piron, 1976, Corollary 3.311). I f H is a complex Hilbert space of dimension at least 3, then every isomorphism of the propositional system of all closed subspaces of H is induced by a unitary or antiunitary operator. The resemblance with Uhlhorn's version of Wigner's Theorem is highlighted by: PROPOSITION 14 ([Piron, 1976, Theorem 2.461). A bijective mapping of the set of all atoms of a propositional system L1 onto the set of all atoms of a propositional system L2 which preserves orthogonality of atoms in both directions can be uniquely extended to a n isomorphism of Ll onto La. 7 GENERALIZATIONS TO INDEFINITE INNER PRODUCT SPACES

7.1 Generalizations of the classical version of Wigner's theorem Let H be a vector space over K = R or @ together with a form (., .) : H x H 4 K which is bilinear and symmetric in the real case and sesquilinear and hermitian in the complex case. An element x E H is called positive if (x, x) > 0 and negative if (x, x) < 0. If H contains positive as well as negative elements, we say that H is an indefinite inner product space or an indefinite metric space, see [~ogntir,19741 for information about general indefinite inner product spaces. Any indefinite inner product space always contains nonzero isotropic elements, i.e. elements x # 0 such that (x, x) = 0. This subject appeared in a paper by Dirac dealing with quantum field theory. in recent decades indefinite inner product spaces proved useful for certain physical problems as well as mathematical questions, see for example the Introduction in [Bracci et al., 19751. If H is a Hilbert space with the product (., .) then any linear mapping a : H 4 H induces a new inner product on H denoted by (., .), and defined by the formula

452

Georges Chevalier

This product allows, in general, to define a structure of indefinite inner product space on H. The operator a is called the metric operator and certain hypotheses on a such as bounded, self-adjoint, invertible ,... are required in order to obtain interesting results about the new product. Many proofs in indefinite inner product spaces obtained in the tradition1 way don't make use of specific properties of this structure, as for example the existence of nonzero isotropic elements, and thus are also valid in Hilbert spaces by choosing a = lH. The definition of a ray in an indefinite inner product space is the same as that in a Hilbert space but the scalar product and orthogonality of rays are now defined by means of the indefinite inner product (., .),. A bijective transformation of the set [HI of all rays which preserve the new scalar product of rays or the orthogonality of rays in both directions will be called a a-symmetry or a I,-symmetry respectively. The first paper [Bracci et al., 19751 dealing with the generalization of Wigner's theorem to indefinite inner product spaces is one of the most interesting on the subject. In its remarkable introduction the authors give numerous references highlighting the imoportance of indefinite inner product spaces in physics, in particular in quantum field theory, and on the other hand references concerning the purely mathematical development of the theory. They also justify their mathematical hypotheses by physical arguments. i al., 19751. In this paper, the metric operator a is Let's summarize [ ~ r a c cet assumed to be a self-adjoint operator with a bounded inverse and Proposition 1 states that, without loss of generality, one can assumes that a2= 1. A particularity of the paper, justified by physical arguments, is that a-symmetries are not defined on the whole set of rays, and therefore a precise definition of some classes of operators with an analogous property is necessary. i al., 19751). Let H be an indefinite inner product DEFINITION 15 ( [ ~ r a c cet space defined by means of a Hilbert space and linear operator a : H --+ H . An operator U with dense domain Du and dense range such that

is called a a-unitary operator. i al., 1975, Proposition 21). A a-unitary operator is PROPOSITION 16 ( [ ~ r a c c et linear having inverse which is a a-unitary operator and it is closable. If in the previous definition, Equation (13) is replaced by (U(x), U(y)), = (Y,x)U, (U(x);U(y))u = - ( x , Y ) ~or (U(x),U(y))u = - ( Y , X ) ~one obtains the definition of, respectively, a-antiunitary operators, a-pseudo unitary operators and a-pseudo antiunitary operators. Since (U(x), U(x)) = -(x, a ) for x # 0 is impossible in a Hilbert space, a-pseudo unitary operators and a-pseudo antiunitary operators do not exist in Hilbert spaces. In indefinite inner product spaces, they only exist if the eigenvalues 1 and -1 of the metric operator a have the same multiplicity. For a-antiunitary operators, a-pseudo unitary operators and a-pseudo antiunitary operators a result similar to Proposition 16 exists, with the difference that a-antiunitary and a-pseudo antiunitary operators are antilinear.

Wigner's Theorem and its Generalizations

453

As we shall see below the big difference between the definite and the indefinite case is the presence of a-pseudo unitary operators and a-pseudo antiunitary operators in the indefinite case in the statement of Wigner's theorem. i al., 19751). Let H be an indefinite inner product PROPOSITION 17 ( [ ~ r a c cet space defined by means of a Hilbert space H and a linear operator a : H -+ H which i s self-adjoint with a bounded inverse. Let T be a bijective mapping defined o n a set R of rays of H onto a set R' of rays such that 1. the set D of vectors belonging to the rays of R and the set D' of vectors belonging to the rays of R' are dense linear subspaces in H;

2. For any rays

[XI,

[y] E R,

There exists an operator U : D --t D' such that, for any x E D , U(x) E T([x]). The operator U i s either a-unitary or a-antiunitary or a-pseudo unitary or a-pseudo antiunitary. In [2000a],Molnk proved a similar result without the hypothesis of self-adjointness for a , see Section 8.2.

7.2

Genemlixations of Uhlhorn's version of Wigner's theorem

The first generalization of Uhlhorn's version of Wigner's theorem to indefinite inner product space appeared in [van der Broek, 1984al. In this paper, the author considers an n-dimensional complex Hilbert space H , n 2 3, and a metric operator a which is self-adjoint and bijective. If T is a I,-symmetry then, as in [Bracci et al., 19751, T is induced by an operator U which is either a-unitary or a-antiunitary or a-pseudo unitary or a-pseudo antiunitary ([van der Broek, 1984a, Theorem 11 or [van der Broek, 1984bl). The main differences which distinguishes this result i al., 19751 are that T is a 1,-symmetry defined on the whole of [HI from [ ~ r a c cet and 3 5 dimH < ca. The author of [van der Broek, 1984a] applies this result to representations of groups in indefinite inner product spaces in [van der Broek, 1984bl In his generalization of the classical version of Wigner's theorem to indefinite inner product space Molnk removes self-adjointness of the metric operator from his hypotheses and proves in [Molnir, 20021 the following generalization of Uhlhorn's version of Wigner's theorem to indefinite inner product spaces. PROPOSITION 18 ([Molnir, 2002, Corollary 21). Let H be a (real or complex) Hilbert space of dimension not less than 3 and let a : H -+ H be a n invertible operator. Suppose that T : [HI + [HI i s a 1,-symmetry. If H i s real then T i s induced by an invertible linear operator U o n H. Similarly, if H i s complex then T i s induced by an invertible linear or antilinear operator U on H . The operator U inducing T i s unique up to multiplication by a scalar.

454

George Chevalier

The invertible linear operator U : H + H induces an I,-symmetry if an only

if (U(X),U(Y))U= c(x, Y ) U

(x, Y E H )

holds for some constant c E R in the real case and c E C in the complez case. If H is complex, then the antilinear operator U : H + H induces an I,symmetry if and only if

holds for some constant d E (C. Here, a* denotes the adjoint of a This result is a corollary of the main theorem of [ ~ o l n ~20021, r, which describes the form of all bijective transformations @ of the set Il(X) of all rank-one idempotents on a Banach space X such that

Its proof is based on a result of Ovchinnikov [1993]describing the automorphisms of the poset of all idempotents on a separable Hilbert H space by means of automorphisms and anti-automorphisms of the lattices of all closed subspaces of H. REMARK. If a is a self-adjoint operator then, for any z E H, (z, z), = (o(z), z) is real and thus the constant d in the previous proposition as well. Let w = 1 or w = and define V : H --+ H by V(x) = -U(x). The operator V generates W the same I,-symmetry as U, is linear or antilinear if U is linear or antilinear respectively and (V(x), V ( Y ) ) = ~ f(x, Y),. 8 SOME OTHER GENERALIZATIONS

8.1

The case of Hilbert modules

Roughly speaking, a Hilbert module is a Hilbert space in which a C*-algebra replaces the complex field. More precisely, let A be a C*-algebra and let 7-1 be a left A-module with a product [.,.] : 'FI x 'FI + A. The module 7-1 is called an A-Hilbert module or a Hilbert C*-module over A if:

2. [af, 91 = a[f, gl ;

3. [g,f l = [f,gl*; 4. [f,f ] 2 0 and [f,f ] = 0

* f = 0;

Wigner's Theorem and its Generalizations

5. 7-1 is complete with respect to the norm f

[f,f

11 4

The concept of Hilbert module is due to I. Kaplansky and in its full generality to Paschke [1973]. In [Molnbr, 19981 and [Molnk, 19991, L. Molnbr proves some generalizations of Wigner's theorem to Hilbert modules using a new algebraic approach and, in particular, results from ring theory. If, as usual in C*-algebras, 1x1 denotes the , can be stated unique positive square root of xx*, the main result of [ ~ o l n b r19991 as follows. PROPOSITION 19 ( [ ~ o l n b r19991). , Let 7-1 be a Hilbert C*-module over the C*algebra A = Mn(C) of all n x n complex matrices with n > 1. Let T : 7-i + 7-1 be a mapping with the property that:

There exists a linear mapping U, called an A-isometry, and cp : 7-1 + C such that:

The absence of A-anti-isometries is a consequence of the noncommutativity of A. In [Bakit and Guljas, 20021, the C*-algebra A = Mn(C) of all n x n complex matrices is replaced by the C*-algebra K ( H ) of all compact operators on a Hilbert space and a result similar to the above proposition is proved. Since the C*-algebra A = Mn(C) of all n x n complex matrices can be considered as the C*-algebra of all compact operators on Cn, this result generalizes Proposition 19.

8.2

Genemlization to pairs of symmetries

Using, as in [Molnbr, 19981 and [Molnk, 19991, a new algebraic approach Molnbr, in [2000a], first proved a generalization of Wigner's Theorem for pairs of symmetries. PROPOSITION 20 ([Molnbr, 2000a, Theorem I]). Let H be a complex Hilbert space of dimension at least 3 and let S and T be two bijections of [HI with the property that:

Then there are invertible either both linear or both antilinear operators U, V : H -t H such that V = U*-I and

Georges Chevalier

456

The proof is not very simple. It uses Gleason's theorem and one of its variations due to A. Dvureeenskij [1993, Chapter 31 and also a result of Jacobson and Rickart stating that any Jordan homomorphiim of a local matrix ring is the sum of a homomorphism and an anti-homomorphism. As a corollary, the author obtains the following generalization of Wigner's theorem to a-symmetries in indefinite inner product spaces. Note that in this proposition, the metric operator a is not assumed to be self-adjoint. PROPOSITION 21 ([Molndr, 2000a, Corollary 21). Let H be a complex Hilbert space with dimH 2 3 and let a : H + H be a n invertible operator. For any a-symmetry T on H there exists a n invertible either linear or anti-linear operator U o n H with U*aU = €afor some scalar E E (C of modulus 1 such that

We remark that if E = 1 then U is either a a-unitary or an a-anti-unitary operator and if 6 = -1, U is either a a-pseudo unitary or a a-pseudo antiunitary operator since in the latter case

8.3 Preservation of angles Let [q5] and [cp] be two rays generated by the unit vectors q5 and cp. The scalar product ([$I, [cp]) = I(q5, cp)l can be interpreted as the cosine of the angle between the rays [I$] and [cp] and so Wigner's theorem states that any bijection of rays which preserves angles of rays in both directions is generated by a unitary or antiunitary operator. This interpretation has led L. Molndr to investigate bijections between , A first problem is subspaces of a Hilbert space preserving angles ( [ ~ o l n d r20011). to make a choice among several possible definitions of the angle of two subspaces in a Hilbert space. Molndr holds that the adequate concept is that of principal angles defined as follows. Let F and G be two finite dimensional subspaces of a Hilbert space with 1 5 7r p = d i m F 5 dimG. T h e p principal angles of F and G, 0 5 el 5 ... 5 0 < -, P-2 are defined recursively by means of their cosines:

a

For 1 < k 5 p,

where [xl, . . . ,x , ] ~denotes the subspace orthogonal to the subspace generated by {XI,.

- .,

57%).

Wigner's Theorem and its Generalizations

457

The definition is not easy. Molnir, however, asserts that a characterization of the cos Ole'sis simple: they are the square roots of the eigenvalues of the positive self-adjoint operator PFPGPF counted by their multiplicity. If L(F,G) denotes the system of principal angles between F and G, then L(Fl, GI) = L(F2,G2) if and only if the operators PFlPGIPF,and PF2PG2 PF2 are unitarily equivalent, that is there exists a unitary operator U such that PF~ P GPF~ ~ = UPFP ~ GPF~ ~ u*. For infinite dimensional subspaces the definition of the principal angles is more difficult. But we still have that the unitary equivalence of PFlPGlPFland PF2PG2 PF2 characterizes the system of principal angles, i.e. L(Fl, GI) = L(F2,G2). PROPOSITION 22 ( [ ~ o l n i r2001, , Main he or em]). Let H be a real or complex Hilbert space with dim H 2 n, n E N. Let Pn(H) and P,(H) if dimH = ca be the sets of all rank-n projections and all infinite rank projections. Suppose that 4 : Pn(H) 4 Pn(H) i s a transformation with the property that

dim H Ifn=l orn#then there exists a linear or conjugate-linear isometry 2 V : H + H such that 4(P) = VPV* ( P € Pn(H)). If H i s infinite dimensional and if the surjective transformation 4 : P, satisfies L(4(P), 4(Q)) = L(P, Q) then there exists a unitary or antiunitary operator U : H 4 H , such that

4(P)

4

P,

UPU* ( P € Pw(H)).

dimH I f n = -then the transformation P + 1 -P preserves angles but cannot be 2 written in the form of the above proposition. Thus this case is really exceptional, and Molnar hopes that in this case the only transformations preserving angles are P + P and P -t 1 - P. This problem is open. In [Aerts and Daubechies, 19831, another notion of angle is considered, corresponding to the first principal angle in the previous one. It is proved that a morphism from the orthocomplemented lattice of all subspaces of a complex Hilbert space H (dim H 2 3) to another preserves the angles of lines.

8.4

Generalization to type 11factors and Banach spaces

Let H be a Hilbert space. The usual trace is a mapping t r defined on the cone of positive operators B+ (a) of the algebra B,(H) of all self-adjoint operators of H with values in the extended positive reals [0,+m]. This mapping satisfies:

Georges Chevalier

Traces on a von Neumann algebra are, as above, defined by the properties I., 2., 3.' and a trace r on the cone M+ of all positive elements of a von Neumann algebra M is said to be a

faithful if x

> 0 implies r ( x ) > 0;

semifinite if for any nonzero x E M+ there exists a nonzero y 5 x such that r ( y ) < +m: normal if

SUP xi) = SUP r ( x i ) for any bounded increasing net

{xi) E M+.

Type I and type I1 factors have a common characterization in terms of traces: a factor is type I or type I1 if and only if it admits a faithful semifinite normal trace.

With the help of Corollary 5, Wigner's theorem can be stated as follows: if T is a bijection of the set of all rank-one projection defined on a Hilbert space H such that t r ( T ( P ) T ( Q ) )= t r ( P Q ) then T can be extended to a *-automorphism or a *-antiautomorphism of B ( H ) . In [Moln&r,2000b], the corresponding result for Type I1 factors is stated as follows. PROPOSITION 23 ( [ ~ o l n k2000b, , he or em]). Let p be a faithful normal semifinite trace on a type II factor M. If T is a bijective transformation of the set of all nonzero finite projections for which

then there is either a linear *-automorphism or a linear *-antiautomorphism

M which extends T .

of

The proof is not easy using deep results from algebra and functional analysis. In [MolnLr, 2000c], Wigner's theorem is generalized to Banach spaces. This generalization is based on formulation of Wigner's theorem given in Corollary 5. There, the Hilbert space H is replaced by a real or complex Banach space X and projections are replaced by rank-one idempotents. Let us denote by X' the topological dual of X and by A' : X' + X' the adjoint of the bounded linear operator A : X + X and by I l ( X ) the set of all rank-one idempotents on X . Wigner's theorem for Banach spaces reads as follows.

Wigner's Theorem and its Generalizations

PROPOSITION 24 ( [ ~ o l n b r2000c, , Theorem 11). Let bijective mapping for which

a1 : Il(X) + Il(X)

459

be a

There exists a bijective bounded operator A : X + X such that

or there exits a bounded biiective operator B : X' + X such that

For the proof Molnar extends to the operator algebra F ( X ) of all finite rank operators and applies a result of [Omladie and ~ e m r l ,19931 related to additive mappings on F(X). The usual version of Wigner's theorem is deduced as a corollary and, in the second part of the paper, a Wigner-type result for matrix algebras over fields of , Theorem 21. characteristic different from 2 is proved [ ~ o l n L r2000c,

8.5 Continuous solutions of 1 (x,y ) 1 = (IS(x),S ( y )1) In the last section of [ ~ a t z 1996], , J. Ratz studies the continuous solutions of the equation I (x, y) 1 = I (S(x), S(y)) 1 where S is a mapping defined on a real or complex inner product space. Since in the usual version of Wigner's theorem the unitary or antiunitary mapping U is defined up to a phase factor, it would seem that the continuous solutions of the equation l(x, y) I = I (S(x), S(y))I are of the form S = WU for a continuous phase factor a. The precise result, however, reads as follows. , Theorem 13, (b) and (c)]). Let H and H' two PROPOSITION 25 ( [ ~ a t z1996, 2 and S : H 4 H' a inner product spaces over IK = W or cG with dimH continuous solution of 1(x,y) 1 = I (S(x), S(y)) 1.

>

I . If

IK = R then S is a linear isometry;

2. If K = cG then there exists an isometry or an anti-isometry U : H 4 H' and f : H + R such that cos of and sin of are continuous on H - (0) and S(x) = eif( x ) ~ ( x ) (x E H). Conversely, all these mappings are continuous solutions of I(x, y)l I(s(x)ls(~>>l-

=

Georges Chevalier

9 QUATERNIONIC HILBERT SPACES

9.1

T h e definite case

The field of quaternions, denoted by W, is a non-commutative field which is also a four dimensional vector space over R. Any quaternion q may be represented in the form q = a bil ciz dig (a, b, c) E R3,

+ + +

with

:i

= -1 and

iris

..

= -zsz,

= it

for any even permutation (r,s, t) of (1,2,3). The set of all quaternions q with b = c = d = 0 is a subfield of W isomorphic to R. The conjugate i j of q is defined by i j = a - bil - cia - dig and the modulus of q b2 c2 d2 = (qq)+. is 191 = Ja2 The center of W, that is the set of quaternions which commute with any quaternion, is W and for any automorphism f of W there exists qo E W, = 1, such that, for any q E W,f (q) = qoqq;l. In other words, any automorphism of W is inner. We remark that for any x E W and any automorphism f , f (x) = x. Let H be is a vector space over W endowed with a definite sesquilinear Hermitian form with respect to conjugation. The pair (H, (., .)) is called a quaternionic Hilbert space if, for any x E H, (x, x) 2 0 and if H is complete for the topology defined by the norm 11 x !I= (x, x);.

+ + +

Quantum mechanics using quaternionic Hilbert spaces has been introduced in et a l , 19621 and also in [E'inkelstein et al., 19591. The first quaternionic version of Wigner's Theorem appeared in '[Uhlhorn, 19621 in its Uhlhorn formulation and a generalization of the classical version of Wigner's theorem to quaternionic Hilbert spaces is established in [Sharma and Almeida, 1990al. There are only two differences with the complex case.

inkel el stein

1. Symmetries and I-symmetries are induced by unitary operators only. This result is linked to the form of the automorphisms of W and can be compared to a result of quaternionic projective geometry: any isomorphism of the lattice of all subspaces of a quaternionic vector space is generated by a linear bijection, 2. Wigner's theorem is false in the two-dimensional case even for symmetries. A counterexample is given in [Bargmann, 19641 and the general form of counterexamples is in [Sharma and Almeida, 1990a].

9.2

Theindefinitecase

In [Molnbr, 20021, the author suggested to generalize his main result to quaternionic Hilbert spaces. This was done in [~emrl,20031 in the definite case and as

Wigner's Theorem and its Generalizations

46 1

well as the indefinite case. As in [ ~ o l n k20021, , the first result of [~emrl,20031 is concerned with bijective transformations of the set of all rank-one idempotent operators which preserve zero products. PROPOSITION 26 ([~emrl,2003, Theorem I]). Let I ( H ) be the set of all rankone idempotent operators of a quaternionic Hilbert space H with dimH 3. If : I ( H ) + I ( H ) is a bijective transformation satisfying

>

TS = 0 if and only if +(T)+(S) = 0 (T, S E I(H)). Then

@(T)= ATA-I

( T E I(H))

where A : H + H i s a invertible semilinear operator. This proposition allows to prove the following generalization of Wigner's theorem to indefinite quaternionic inner product spaces. PROPOSITION 27 ([~emrl,2003, Theorem 21). Let H be a quaternionic Hilbert space with dim H 2 3 and let a : H -+ H be an invertible operator. If T i s a I,-symmetry then there exist a nonzero c E W and a bounded semilinear bijective operator U : H + H such that T([x]) = [U(x)]for every nonzero x E W and

where a : W + ]HI i s the automorphism of the field W corresponding t o the semilinear operator U. REMARK. Assume that a = l~ and let d be the quaternion such that a(%) = d-lxd with Id1 = 1. The real number c in (14) is positive by 11 U(x) [I2= c 11 x 112 and if V : H + H is defined by V(x)

=

d

-U(x)

fi

then V is an operator satisfying

We have got the quaternionic version of Uhlhorn's generalization of Wigner's theorem as in [Sharma and Almeida, 1990al. 10 A TOPOLOGICAL AND LATTICE APPROACH In order to obtain a very general form of Wigner's theorem in its Uhlhorn version, we will, in this Section, replace the Hilbert space H by a topological vector space E over a field K. The first problem is to define an orthogonality relation on the set of all lines of E . In general, orthogonality relations on a lattice of subspaces are defined by means of non degenerate bilinear forms and usually no natural bilinear form is available on E x E . On the other hand, let E* be the algebraic dual space of E

462

Georges Chevalier

formed by all linear functionals on E. There always exists a natural non degenerate bilinear form on E x E*, namely the mapping B : E x E* t K defined by B(x, y) = y(x), x E E, y E E*. Since E is a topological space, closed subspaces seem more convenient than general subspaces and this condition forces us to replace E* by E', the topological dual of E formed by all continuous liiear functionals on E. But now the restriction of the bilinear form B to E x E' is not necessarily non degenerate. We therefore consider only pairs (E, E') which are pairs of dual spaces in the sense of Mackey [I9451 or DieudonnC [1942]. If (E, F ) is a pair of dual spaces then the lattice of all closed subspaces of E is an irreducible complete DAC-lattice and such lattices appear as the natural setting of the lattice part of this study. In the first part of this Section, we will specify the definitions and the main properties of pairs of dual spaces and DAC-lattices. The second part is devoted to the lattice tools necessary to the generalization of Wigner's theorem. In particular we will prove a generalization of the First Fundamental Theorem of projective geometry to lattices of closed subspaces. As a consequence of the previous results, different Wigner-type theorems are proved in the last part. In the following, we will assume that the dimensions of all vector spaces is not less than 3 and the heights of all the lattices are not less than 4.

1 0 1 DAC-lattices and pairs of dual spaces An AC-lattice is an atomistic lattice with the covering property: if p is an atom a n d a A p = O t h e n a a a v p , that i s a < x < a V p i m p l i e s a = x o r a V p = x . In general, At(L) will denote the set of all atoms of lattice L and if L* is the dual lattice of L then At(L*) is also the set of all coatoms of L. If L and and its dual lattice L* are AC-lattices, L is called a DAC-lattice [Maeda and Maeda, 19701. Irreducible complete DAC-lattices of heights 2 4 are representable by lattices of closed subspaces and many lattices of subspaces are DAC-lattices. Let us specify this last assertion. Let K be a field, E a left vector space over K , F a right vector space over K. If there exists a non degenerate bilinear form B on E x F, we say that (E, F ) is a pair of dual spaces [DieudonnC, 19421. Since the form is non degenerate, F can be interpreted as a subspace of the algebraic dual E* of E and E as a subspace of F*. This interpretation allows one to write, for any x E E and any y E F , X(Y)= Y(Z)= B(x, Y). For example, if E is a locally convex space and E' its topological dual space then (E,E') is naturally a pair of dual spaces with B(x, y) = y (x) [Kothe, 1969, page 2341. For a subspace A of E, we put

A~ = {y E F I B(x, y) = 0 for every x E A).

Wigner's Theorem and its Generalizations

Similarly, let B I = {x E E

I B(x, y) = 0 for every y E B )

for every subspace B of F . A subspace A of E is called F-closed if A = ALL and the set of all F-closed subspaces, denoted by LF(E) and ordered by set-inclusion, is a complete irreducible DAC-lattice [ ~ a e d aand Maeda, 1970, Theorem 33.41. Conversely, for any irreducible complete DAC-lattice L of height 2 4, there exists a pair (E, F) of dual spaces such that L is isomorphic to the lattice of all F-closed subspaces of E [Maeda and Maeda, 1970, Theorem 33.71, [Kothe, 1969, §10.3]. The set LE(F) of all E-closed subspaces of F is similarly defined and is also a DAC-lattice. The two DAC-lattices LF(E) and LE(F) are dual isomorphic by the mapping A + AL [ ~ a e d aand Maeda, 1970, Theorem 33.41 and an element X of LF(E) and an element Y of LE(F) are said to be orthogonal if X C YI (Equivalently, Y C x L ) and we write X IY. Let (E, F ) be a pair of dual spaces. The linear weak topology on E, denoted by a ( E , F ) , is the linear topology defined by taking {GL I G C F, dim G < m) as a basis of neighborhoods of 0. If F is interpreted as a subspace of the algebraic dual of E then a subbasis of neighborhoods of 0 consists of the kernels of elements of F . The linear weak topology on F, denoted by a(F,E), is defined in the same way. The space F can be interpreted as the topological dual of E for the a(E,F ) topology and E as the topological dual of F for the a(F,E) topology. Equipped with their linear weak topologies, E and F are topological vector spaces [Kothe, 1969, ~10.31if the topology on K is discrete. Moreover, for a subspace G c E, we have ?? = GIIand thus to be a closed subspace in E is an unambiguous notion. If K = W or C, this result generalizes to any pair (E, F ) and any locally convex topology over E when F is the dual of E for this topology [Kothe, 1969, $20.31.

10.2

The adjoint of a semi-linear map

For a future generalization of the First Fundamental Theorem of projective geometry, it is necessary to make precise the concept of the adjoint of a semilinear mapping. Let (E, F ) be a pair of dual spaces and f : E + E a r-linear mapping with respect to an automorphism r of K. If y is an element of E* then the mapping x E E 4 ~ - l ( ~ (x))) ( f belongs to E*. Let us define f * : E* --t E* by f*(y)(x) = ~ - l ( ~ ( f ( x ) for ) ) any y E E* and any x E E. The mapping f * is r-l-linear and will be called the adjoint of f . Assume that f is weakly continuous. If y E F c E* then the mapping x E E + ~ - ' ( ~ ((x))) f is weakly continuous and therefore f*(y) E F. The restriction of f * to F is weakly continuous [Chevalier, 2005b] and in what follows, i f f : E t E is a weakly continuous r-linear mapping then f * will always mean the restriction of f* to F C E* and thus f ** is a mapping from E to E.

464

Georges Chevalier

Now, let us consider a clmed subspace X of E. For any x E X and any y E F , y ( f ( x ) ) = 0 is equivalent to f * ( Y )( x ) = 0 and so, as for linear mappings, f *-l(xL) = f ( X ) I for any X E L F ( E ) . Others results about adjoints are: f** = f for a weakly continuous semilinear mapping and f*-l = f-l* if f is a weakly continuous semilinear bijection with a weakly continuous inverse [Chevalier, 2005bl.

10.3 The lattice tools for a lattice approach to Wigner's theorem The first propmition is comparable to Proposition 14. PROPOSITION 28 ([chevalier, 2005b, Proposition 11). Let L be a complete D A C lattice. I f f is an automorphism of the poset A t ( L ) U A t ( L * ) then f extends to a n automorphism B of the lattice L. Idea of the proof. For two families of atoms (pi)iEr and ( q j )j E

of L

and an extension B of f to L can be defined by B ( 0 ) = 0 and ,(z) =

V f (pi) if iEI

0 # x = V p i , . Using (l5), it is easy to check that B is an automorphism of the iEI

DAC-lattice L. The following result generalizes the First Fundamental Theorem of projective , to lattices of closed subspaces. In its statement a bicontingeometry [ ~ a e r19521 uous bijection is meant to be a continuous bijection with a continuous inverse. PROPOSITION 29 (Chevalier [2005a; 2005bl). Let ( E l ,F l ) and (E2,F 2 ) be two pairs of dual spaces over the fields Kl and Kz.

+

1. If there exists a n isomorphism of the lattice LFl(E1)onto the lattice LFz (E2)then Kl and Kz are isomorphic fields and there exists a bicontinuous semilinear bijection s : El H E2 such that, for every Fl-closed subspace M of El, + ( M ) = s ( M ) . If a bicontinuous r-linear bijection s and a bacontinuous TI-linear bijection st generate the same automorphism then there exists k E K 2 such that T I = k r k k l and s f = ks.

+

2. If that K1 and Kz are isomorphic fields then, for every semilinear bijection s : El H E2, the following statements are equivalent: (a) The bijection s i s bicontinuous. (b) H E LF,( E l )I+ s ( H ) is a bijection from the set of all Fl-closed hyperplanes of El onto the set of all F2-closed hyperplanes of E2. (c) M E LFl ( E l )H s ( M ) is a n isomorphism from the lattice LFl( E l ) into

-

L F 2 (E2)

Wigner's Theorem and its Generalizations

465

+

Idea of the proof. The mapping is an order isomorphism of the poset of all finite dimensional subspaces of El into the poset of all finite dimensional subspaces of Ez. This isomorphism extends to an isomorphism cp of the lattice of all subspaces of El into the lattice of all subspaces of E2by p(N) = U{$(M)

I M c N,

dim M

< m)

and cp extends $. The First Fundamental Theorem of projective geometry then concludes the proof of 1. The bijection s maps the set of all hyperplanes of El bijectively into the set of all hyperplanes of E2 and , if s is bicontinuous, closed hyperplanes of El correspond with closed hyperplanes of E2. Thus (a) + (b) holds true. The proof of (b) + (c) uses the fact that in a DAC-lattice LF(E), any element is an intersection of closed hyperplanes. Since the family of all closed hyperplanes is a O-neighborhood subbasis for the linear weak topology (c) +- (a) is clear.

10.4

Wigner-type theorems

PROPOSITION 30 (A Wigner-type theorem for DAC-lattices). Let L be an irreducible complete DAC-lattice and f an automorphism of the poset At(L) UAt(L*). If L is representable as the lattice LF(E) of all F-closed subspaces of a pair of dual spaces (E, F ) then f extends to an automorphism q5 of LF(E) and there exists a bicontinuous semilinear bijection s : E -+ E such that @(M)= s(M) for all M E LF(E). Proof. Use Propositions 28 and 29. REMARK. Let L be the lattice of all subspaces of a vector space E. If f is an automorphism of At(L) U At(L*) (informally speaking, f preserves in both directions inclusion of lines in hyperplanes) then f extends to an automorphism of L and there exists a semilinear bijection s : E -t E such that, for any subspace

x,f (XI = 4x1. Let (E, F ) be a pair of dual spaces. If f : At (LF (E)) u A ~ ( L E (F)) + At (LF (E)) U At(LE(F)) is at the same time a bijection of A~(LF(E))and a bijection of At(LE(F)) such that, for any p E At(LF(E)) and any q E At(LE(F)),

then f is called an I-symmetry over (E, F ) . PROPOSITION 31 (A Wigner-type theorem for a pair of dual spaces). Let f be an I-symmetry over a pair (E, F ) of dual spaces. There exists a bicontinuous semilinear bijection s : E -+ E such that:

Georges Chevalier

2. for any q E A ~ ( L E ( F )f) (q) , = s*-'(q). Proof. As M E L F ( E )-+ M-'- E L E ( F )is an anti-isomorphism of lattices, we can define a bijection fl of At(LF(E)*)by fl(P) = f ( p L ) l for any P E At(LF(E)*). Let g be the extension of fl to A t ( L F ( E ) U ) At(LF(E)*)which agrees with f on At(LF( E ) ) .If p E At(LF( E ) )and P E A ~ ( L F ( E ) *we ) , have p 5 P if and only if p IPL which is also equivalent to f (p) I f ( p L )= g ( ~ ) that L is g(p) 5 g(P). By Proposition 28, g extends to an automorphism G of the lattice L F ( E )and by using Proposition 30 there exists a bicontinuous semilinear bijection s such that, for every F-closed subspace M , G ( M )= s ( M ) . In particular, for every atom p of LF( E ) ,S ( P ) = G(P)= g ( p ) = f b). Let q E At(LE(F)).We have :

10.5 Examples = W or C is weakly continuous if and only if f is continuous with respect to the linear weak topology u ( E ,El) [Kothe, 1969, 20.41. If K = W then a semi- linear mapping is linear since the identity is the only automorphism of W and we have the following version of Wigner's theorem.

A linear mapping f , defined on a locally convex space E over K

COROLLARY 32. Let E be a real locally convex space and E' its dual. I f f is an I-symmetry over the dual pair ( E ,E') then there exists a weakly bicontinuozls linear bijection s : E + E such that

for any p E A ~ ( L E ~ ( Ef )b) ), =sb), for any q E A ~ ( L E ( E ' )f) (q) , = s*-'(q). If E is metrizable then s is continuous. For the last claim of this corollary we have used the fact that weakly continuous linear mappings between metrizable spaces are continuous [Schaefer, 1964, Chapter IV, 3.4 and 7.41. If K = C then the automorphism r associated to the semi-linear bijection s of Proposition 31 cannot be continuous (In a locally convex space over a field K , the topology on K is not the discrete one but is defined by means of the modulus) and an extra hypothesis seems necessary to obtain a version of Wigner's theorem close to the classical one.

Wigper's Theorem and its Generalizations

467

COROLLARY 33. Let E be a n infinite-dimensional complex n o w e d space and f a n I-symmetry over the dual pair ( E lE'). There exists a linear or conjugate linear bijection s : E + E which is bicontinuous for the n o r m topology and such that:

for any q E A ~ ( L E ( E ' f)(, q ) = s*-'(q). Proof. Let s be the semi-linear bijection obtained by using Proposition 31. Since s is continuous for the weak linear topology, s carries orthogonally closed hyperplanes to orthogonally closed hyperplanes. But orthogonally closed subspaces of E agree with topologically closed subspaces he, 1969, 520, 3 (2)] and by using a result of [Kakutani and Mackey, 19461 or illmo more and Longstaff, 19841, Lemma 2, s is either linear or conjugate linear. A linear mapping on a metrizable space E is continuous if and only if this mapping is continuous for the linear weak topology a ( E ,E') and the generalization of this result to a conjugate linear mapping is easy. Thus, s is continuous and, by using a similar argument, s-l is also continuous. REMARKS.

1. In [2002], L. MolnLr proved the same result for complex Banach spaces. 2. If E is a finite-dimensional complex normed space and s : E + E is any T-linear bijection then one can define an I-symmetry on ( E lE') by f (p) = s(p) if p E &(LEI ( E ) ) and f ( q ) = s * - ' ( ~ ) if q E,A ~ ( L E ( E ' )If. the automorphism T of C is neither the identity nor the conjugation then s is not continuous.

The following corollary is the classical version of Wigner's Theorem in its Uhlhorn version. What is of interest here is only its proof, which uses the previous results and especially the Wigner-type theorem for pairs of dual spaces. COROLLARY 34. Let H be a Hzlbert space over K = B or cG (dim H 2 3) and T an I - s y m m e t r y o n [HI. There exists a semilinear mapping r : H + H such that: 1. for any ray p E [HI, r(p) = T ( p ) ,

2. if K = B,r i s a unitary operator, 3. If K = C, r is either a unitary or an antiunitary operator.

T h e mapping T extends to a n automorphism cP of the orthomodular lattice of all closed subspaces of H

Georges Chevalier

468

Proof. Since H is a Hilbert space, the correspondence 6 which associates to every y E H the continuous functional 6(y) : x + (x, y) is an isomorphism from H onto its dual H' in the real case and an anti-isomorphism in the complex case. This mapping generates an isomorphism, denoted by the same symbol, from the lattice of all closed subspaces of H onto the lattice of all closed subspaces of its dual. with two different Remark that for (p,q) E [HI x [H'], p Iq ++ p I meanings for the ~ r t h o g o n ~ relations. ty Let S be the I-symmetry on (H, H') defined for (p, q) E [HI x [H'] by S(p) = T(p) and S(q) = 6(T(B-'(q))). Assume that H is infinite-dimensional in the complex case. If B is always the canonical bilinear form on the pair (H, H') then for any (5, y) E H x H', B(x, y) = (x, 6-'(y)) and if f f l : H t H denotes the adjoint, defined by means of the inner product of the linear or antilinear operator f , then f f l = 6-' f*6 where f* : H' t H' is the adjoint of f defined in 10.2. By using Corollary 32 or Corollary 33, there exists a linear or antilinear bijective operator s : H + H such that for any p E [HI, s(p) = S(p) = T(p) and for any q E [H'], s*-'(9) = S(q) = O(T(6-'(q))). Therefore, for any p E [HI, T(p) = 8-'(s*-'(6(p))) = SU-'(~). Thus s and su-' generate the same automorphism of the lattice of all subspaces of H and there exists X E K such that s = Xsfl-'. We have ssfl= XIK and therefore X > 0. 1

If r = -s

fi

then, for any ray p, r(p) = T(p) and r-'

= rfl. If

K

=

R, r is a

unitary operator and, in the infinitedimensional complex case, r is a unitary or antiunitary operator. Now, we assume that H is a finite-dimensional complex Hilbert space. If f is a T-linear mapping then, for any y E H, the mapping x E H t 7-I ((f (x), y)) is a linear form and there exists an element f fl(y) E H such that ~ - ' ( ( f (x), y)) = (x, ffl(y)). It is easy to check that the mapping f fl : x E H + ffl(z) E H satisfies f#(y y') = ffl(y) fU(yr) and ffl(Xy) = ~-l(X)f(y) if Y, y' E H, X E C. Moreover, f f l = O-'f*6. As in the infinite-dimensional case, let S be the I-symmetry on (H, H') defined for (p, q) E [HI x [H'] by S(p) = T(p) and S(q) = 6 ( ~ ( 6 - ~ ( q ) ) )If. s : H t H is the T-linear bijection obtained by using Proposition 31 then, as in the infinitedimensional case, there exits a bijection r such that r-' = rfl and r(p) = T(p) for any P E [HI. For any p E C and 0 # x E H, r-'(px) = 7-'(p)r-'(2) and rfl(px) = T-l(jZ)rfl(x). Therefore ~ - ' ( p ) = ~-l(jZ) and thus r-'(IW) = R. The automorphism T is the identity or the conjugation and r a unitary or antiunitary operator. Let Q, be the automorphism of the lattice of all closed subspace generated by the unitary or antiunitary operator r. This automorphism extends T and by using = r , we the relations rfl-'(XI) = T ( x ) ~for a closed subspace X of H and TU-' . mapping Q, is an automorphism of the orthomodular have @ ( X I )= @ ( x ) ~ The lattice of all closed subspaces of H. This last result can be also obtained by using Proposition 14.

+

+

Wigner's Theorem and its Generalizations

11 SOME OTHER SYMMETRY GROUPS

In his books (Wigner [1931; 19591) Wigner postulated the invariance of the transition probabilities between pure states and this hypothesis leads to the symmetry group of a physical system. As in [Cassinelli e t al., 19971 or [Cassinelli e t al., 2004, Chapter 21, one can consider several other objects playing a fundamental role in the Hilbert space formulation of quantum mechanics. Among these there are: 1. The convex set S of all states. This set is represented by the set of all positive trace class operators of trace 1 on H.

2. The orthomodular lattice L of all closed subspaces of H. This lattice r e p resents the lattice of all yesno experiments or propositions on the physical system. 3. The effect algebra E ([~reechieand Foulis, 19951) of all positive operators bounded by the unit operator.

4. The Jordan algebra B, of all bounded self-adjoint operators. In this algebra, AB+BA the product is the Jordan product defined by A o B = and an 2 element of this algebra can be interpreted as a bounded observable. 5. The C*-algebra B of all bounded operators on H. To each of these objects there corresponds a group of automorphisms, i.e. bijective mappings preserving the relevant structures. Let us give three examples.

1. . An automorphism of the orthomodular lattice L is any bijective mapping f : L + L such that for any closed subspaces M and N of H

Any automorphism preserves all joins and meets.

2. If A and B are two positive operators bounded by the unit operator I of H then, in the effect algebra E, the sum A+ B is defined only if A+ B 5 I. If it is the case, it agrees with the usual sum of two operators. An automorphism of E is a bijective mapping f : E + E such that and in this case, f (A+ B) = f (A) f (B). Any automorphism preserves the unit operator and the order ([Cassinelli e t al., 19971).

+

3. An element of Aut(B) is a linear or antilinear bijection satisfying @(AB)= @(A)@(B)and @(A*)= @(A)*,A, B E B. This is not the usual definition of a C*-algebra automorphism.

Georges Chevalier

If Aut(X) is one of the above groups of automorphisms then it can be considered as the group of all transformations of the physical system preserving the physical features associated to X . Thus its elements are the symmetries of the system when the formulation of quantum mechanics is based on the set X of objects. For example, Piron defines a symmetry as an isomorphism of the orthomodular lattices L of all propositions iron, 1976, page 371. The following proposition shows that all these automorphism groups are essentially the same. Roughly speaking, if a transformation of a physical system leaves invariant some convenient physically significant features then all the physically significant features are also invariant. PROPOSITION 35 ([Cassinelli et al., 19971).

1. If dim H 2 2 then all the automorphism groups Aut(P), Aut(B), Aut(B,), Aut(E) and Aut(S) are isomorphic. 2. If dim H 2 3, then the above groups are also isomorphic to Aut(L) and to the group Autl(P) of all bijections of P preserving orthogonality. Shortened proof. In [Cassinelli et al., 19971, the authors prove the existence of five injective group homomorphisms Aut(P)

% Aut(B) % Aut(B,) % Aut(E) 2 Aut(S) 5 Aut(P)

defined as follows. 1. Any automorphism of P (identified here to the set of all rank-one projections) is of the form P + UPU* with U unitary or antiunitary (Corollary 5) and a1 associates to this automorphism the automorphism A E B + UAU* of

B. 2. Since E c B, c B the definitions of Q2 and a3are easy: it suffices for Qa to check that the restriction of an element of Aut(B) to B, belongs to Aut(B,) and similarly for a3. 3. The definition of a4is more difficult. In [Cassinelli et al., 19971, it is proved that, for'any f E Aut(E) and any state p, there exists a state p' such that, for any E E E, tr[p o f -'(E)] = t ~ [o ~E)] ' and

a4is defined by @4(f)(p) = p'

4. Since P c S, it suffices for defining a5to check that the restriction to pure states of a mapping preserving the convex structure of S also preserves transition probabilities between pure states [Cassinelli et al., 1997, Proposition 4.61.

Wigner's Theorem and its Generalizations

471

All the previous homomorphisms are isomorphisms since by a straightforward computation, using the properties of the trace for @4, If dim H 2 3 then, by using Proposition 14, Aut(L) is isomorphic to Autl(P) and Corollary 13 implies that Aut(L) and Aut(P) are isomorphic. REMARKS.

1. If dim H = 2 then Aut(P) is a proper subgroup of Autl(P) [~hlhorn,1962, 5.21, [~assinelliet al., 1997, Example 4.11. This result can be also obtained by a comparison of the cardinals of Aut(P) and Autl(P). Indeed, Card(Aut(P)) = Card(C) and Card(Autl(P)) is the cardinal of the set of all bijections from C to C, that is 2Card". Thus, if dim H = 2, Aut(P) and Autl(P) are dserent and non-isomorphic groups. Without any assumption on the dimension of H, the groups Autl(P) and Aut(L) are isomorphic [Cassinelli et al., 1997, Proposition 4.8 and Corollary 4.31. 2. In [Cassinelli et al., 19971, the previous automorphism groups are all endowed with natural initial topologies and it is proved that there are all homeomorphic topological groups. Each of them is a second countable metrizable topological group where second countable means that the open sets have a denumerable basis. 3. In [1962], Uhlhorn associated to any automorphism f of P a mapping Q(f) : B + B defined by If f is generated by a unitary operator U then Q(f)(A) = UAU*, If f is generated by an antiunitary operator U then Q(f )(A) = UA*U*, AEB. If U is unitary then Q(f) is an automorphism of the C*-algebra B and otherwise, Q(f) is an anti automorphism, that is Q(f)(AB) = Q(f)(B)Q(f)(A). The mapping Q is an isomorphism of the group Aut(S) into the group of all automorphisms or antiautomorphisms of the C*-algebra B. If, as in [~assinelliet al., 19971, one consider Q1(f) defined in the same way for U unitary or antiunitary by Q1(f)(A) = UAU* then Q1is an isomorphism of the group Aut(P) into the group Aut(B) where an element 4 of Aut(B) is defined as a linear or antilinear bijection satisfying 4(AB) = 4(A)+(B) and 4(A*) = +(A)*. In the antiunitary case, such automorphism is neither an automorphism nor an antiautomorphism of the C*-algebra B in the usual sense. 4. The isomorphism of Aut(P), Aut(S) and Aut(B,) is also a consequence of Theorems 2.1, 2.2 and 2.3 in [Simon, 19761. A more elementary study of these groups may be found in [Hunziker, 19721.

Georges Chevalier

12 FROM AUTOMORPHISMS TO THE HAMILTONIAN In this concluding Section we will show how Wigner's theorem permits us to derive the Schrdinger equation for a conservative physical system. While the proof of Wiper's theorem is quite elementary, this derivation requires deep results from functional analysis, so we will outline only its main steps. For more information, see [Jauch, 1968, Chapter 101, [Beltrametti and Cassinelli, 1981, Chapters 6 and 231. For a rigorous derivation and to go deeper in the question see [~argmann, 1970; Simon, 19761 or [Varadarajan, 19851; [Jordan, 1991] is also an interesting reference. First, we will assume that the state of the physical system at time t is completely determined by its state at any time to < t, such a system is called a conservative system. Let us denote by [cpt] the state at time t and by Vt,t~the two-parameter family of transformations of states such that [cptt] = Vt,t~([cpt])if t < t'. If to < tl < tz then

and thus

Now if we assume that Vt,t~depends only on the difference of times r = t' - t then (16) becomes

This hypothesis corresponds to those situations in which there is homogeneity of time and, in the terminology of Markov process, to a stationary Markovian evolution. Another requirement that has natural physical motivations is the continuity of the real function -,(&([PI)>[+I) for every pure states [cp] and [+I. Intuitively, this hypothesis means that small changes in time produce small changes in probabilities. We also assume the reversibiity of the evolution of the system which forces V, t o be invertible with Vyl = V-,. Finally, r 4 V, is a homomorphism from the group (R, +) into a group of bijective transformations of the set of all states. In order to use Wigner's theorem a last hypothesis is necessary: r + V, preserves the transition probabilities, i.e.

for every pure states [cp] and

[$I.

Wigner's Theorem and its Generalizations

473

Now we are in position to apply Wigner's theorem: V, is phase equivalent to a unitary or antiunitary operator U, defined up to a phase factor and considering that U, = U+o U; , U, is in fact a unitary operator. Using Equation (17), we have

and r -+ U, is a projective representation of the additive group of W into the group of all unitary operators of H. By a result of Wigner [1939],(see also [Simon, 1976]), w(r, 7') can be chosen in such a way that w(r, 7') = 1 and Equation (18) becomes

Thus T + U, is a strongly continuous one-parameter unitary group [Reed and Simon, 1972, page 2651. Using Stone's theorem [Reed and Simon, 1972, page 2641, [Simon, 1976, Theorem VIII], we get that there exists a self-adjoint operator A on H such that - -irA. ,-e

In mathematics, the self-adjoint operator A is called the infinitesimal generator of U, and, in physics, the Hamiltonian of the system. It represents the physical quantity called the energy of the system. Now the Schrodinger equation is at hand: if the state of the system at time t is $(t) then +(t)= eVirA4(0)and satisfies the Schrodinger differential equation

+

BIBLIOGRAPHY [Aerts and Daubechies, 19831 D. Aerts and I. Daubechies. Simple proof that the structurepreserving maps between quantum-mechanical propositional systems conserve the angles, Helvetica Physica Acta 56, 1187-1190, 1983. [Baer, 19521 R. Baer. Linear algebra and projective geometry, Academic Press Inc. 1952. [BakiC and GuljaB, 20021 D. BakiC and B. GuljS. Wigner's Theorem i n Hilbert C*-modules over C-algebras of compact operators, Proc. Amer. Math. Soc. 130 ( 8 ) , 2343-2349, 2002. [Bargmann, 19641 V. Bargmann. Note on Wigner's T h w r e m o n symmetry operations, J . Math. Phys. 5 , 862-868, 1964. [Bargmann, 19701 V. Bargmann. O n unitary ray representations of continuous groups, Annals of Math., 59 ( I ) , 1-46, 1954. 1970. [Beltrametti and Cassinelli, 19811 E. Beltrametti and G. Cassinelli. The logic of quantum mechanics, Addison- Wesley Publishing Company, 1981. [Birkhoff, 19671 G. Birkhoff. Lattice theory, Amer. Math. Soc. Colloquium Publications,Volume XXV, 1967. [Birkhoff and von Neumann, 19361 G . Birkhoff and J. von Neumann. The logic of quantum mechanics, Ann. Math., 37, 823-843, 1936. [Bognk, 19741 J . Bognkr. Indefinite Inner product spaces, Springer-Verlag 1974. [Bracci et al., 19751 L. Bracci, G. Morchio and F. Stocchi. Wigner's theorem o n symmetries i n indefinite metric spaces, Cornmun. Math. Phys. 41, 289-299, 1975. [Cassinelli et al., 19971 G. Cassinelli, E. de Vito, P. Lahti and A. Levrero. Symmetry groups i n quantum mechanics and the theorem of Wigner o n the symmetry transformations, Rar. Math. Phys. 9 , 921-941, 1997.

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[Cassinelli et al., 20041 G. Cassinelli, E. de Vito, P. Lahti and A. Levrero. The Theory of Symmetry Actions i n Quantum Mechanics, Springer, 2004. [Chevalier, 2005al G. Chevalier. Automorphisms of a n orthomodular poset of projections, Int. J. Theor. Phys. 44 (7), 985-998, 2005. [Chevalier, 2005b] G. Chevalier. Lattice approach to Wigner-type thwrenw, Int. J. Theor. Phys. 44 ( l l ) , 1903-1913, 2005. [Di?udonnk, 19421 J. DieudonnB. La dualit6 d a m les espaces vectoriels topologiques, Ann. Sci. Ecole Norm. Sup., 59, 107-139, 1942. [Dvure~enskij,19931 A. DvureEenskij. Gleason's Theorem and its applications, Kluwer Academic Publisher, 1993. [Emch and Piron, 19631 G. Emch and C. Piron. Symmetry i n quantum theory, J . Math. Phys. 4 (4), 469-473, 1963. [Fillmore and Longstaff, 19841 P. A. Fillmore and W. E. Longstaff. Zsomorphisnzs of lattices of closed subspaces, Can. J. Math. Vol XXXVI (5), 820-829, 1984. inkel el stein et al., 19591 D. Finkelstein, J. M. Jauch and D. Speiser. Notes on quaternion quant u m mechanics, Cern Theoretical Study Division No 59-7 1-17, No 59-9 1-36, No 59-17 1-28, 1959. [Finkelstein et al., 19621 D. Finkelstein, J. M. Jauch, S. Schiminovich and D. Speiser. Foundations of quaternion quantum mechanics, J. Math. Phys. 3, 207-220, 1962. [Greechie and Foulis, 19951 R. Greechie and D. Foulis. 'Pansition to effect algebras, Int. J . Theor. Phys. 34 (8), 1369-1382, 1995. [Gyory, 20041 M. Gyory. A new proof of Wigner's Theorem, Rep. Math. Phys. 54 (2) 159-167, 2004. [Henley, 20011 E. M. Henley. What do we know about time reversal invariance, Fisika B 10 (3), 161-165, 2001. [Houtappel et al., 19651 R. M. F. Houtappel, H. van Dam and P. Wigner. The conceptual basis and w e of the geometric invariance principles, h v . Mod. Phys. 37, 595-632, 1965. [Hunziker, 19721 W. Hunziker. A note o n symmetry operations in quantum mechanics, Helvetica Physica Acta 45, 233-236, 1972. [Kakutani and Mackey, 19461 S. Kakutani and G. Mackey. Ring and lattice characterizations of wnzplex Hilbert spaces, Bull. Amer. Soc. 52, . 727-733, 1946. [Kalmbach, 19831 G. Kalmbach. OrUlonzodular lattices, Academic Press, 1983. [Keller, 19801 H. A. Keller. Ein nicht-klassischer Hilbertscher m u m , Mathemathische Zeitschrift, 172, 41-49, 1980. [Kothe, 19691 G. Kothe. Topological vector spaces I, Springer-Verlag, 1969. [Jauch, 19681 J. Jauch. Foundations of quantum mechanics, Addison-Wesley Publishing Company, 1968. [Jordan, 19911 T . F. Jordan. Assumptions implying the Schrodinger equation, Amer. J. Phys. 59 (7), 606-608, 1991. [Lomont and Mendelson, 19631 J. S. Lomont and P. Mendelson. The Wigner unitaryantiunitary theorem, Annals of Math. 78 (3), 548-559, 1963. [Mackey, 19451 G. Mackey. O n infinite-dimensional linear spaces, Trans. Amer. Math. Soc. 57, 155-207, 1945. [Maeda and Maeda, 19701 F. Maeda and S. Maeda. Theory of symmetric lattices, SpringerVerlag, 1970. [Molniir, 19961 L. Molnk. Wigner's unitary-antiunitary theorem via Herstein's theorem on Jordan homomorphisnm, J. Nat. Geom. 10, 137-148, 1996. [ ~ o l n r i r19981 , L. Molnir. A n algebraic approach to Wigner's unitary-antiunitary theorem, J . Austral. Math. Soc. 65, 354-369, 1998. [Molniir, 19991 L. Molniir. A generalization of Wigner's unitary-antiunitary theorem to Hilbert modules, J . Math. Phys. 40 ( l l ) , 55445554, 1999. [MolnAr, 2000aI L. Molniir. Genemlization of Wigner's unitary-antiunitary theorem for indejinite inner product spaces, Commun. Math. Phys. 210, p. 785-791, 2000. [Molnrir, 2000bI L. Molnk. Wigner-type thwrem o n symmetry tmnsfomaations in type II factors, Int. J. Thwr. Phys. 39 (6), 1463-1466, 2000. [Molniir, 2000c] L. Molnir. Wigner-type thwrem o n synznletry tmnsfornzations i n Banach spaces, Publ. Math. (Debrecen) 58 , 231-239, 2000.

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[Molnir, 20011 L. M o l n k . !lkansfornaations o n the set of all n-dimensional subspaces of a Hilbert space preserving principal angles, Commun. Math. Phys. 217, 409-421, 2001. [ M o l d r , 20021 L. Molnir. Orthogonality preserving transformations o n indefinite inner product spaces: Generalization of Uhlhorn's version of Wigner's theorem, Journal o f Functional Analysis 194, 248-262, 2002. [OmladiE and Semrl, 19931 M. OmladiE and P. Semrl. Additive mappings preserving operators of rank-one, Linear Algebra Appl. 182, 239-256, 1993. [Ovchinnikov, 19931 P. Ovchinnikov. Automorphwms of the poset of skew projections, Journal o f functional analysis 115, 184189, 1993. [Paschke, 19731 W . Paschke. Inner pmduct spaces over B*-algebras, 'lkans. Amer. Math. Soc. 182, 443-468, 1973. [Pian and Sharma, 19831 J. Pian and C . S. Sharma. Calculw o n wmplez Banach spaces, Inter. J . Theo. Phys. 22, 107-130, 1983. [Piron, 19761 C . Piron. Foundations of quantuna physics, W . A. Benjamin, Reading, Massachussetts, 1976. [Poschadel, 20001 N . Poschadel. A note o n families of observables and a generalization of Wigner's theorem i n c', J . Math. Phys. 41 ( l l ) , 7832-7838, 2000. [Rlitz, 19961 J . Rlitz. O n Wigner's theorem: Remarks, wnaplements, comments, and corollaries, Aequationes Mathematicae 52, 1-9, 1996. [Reed and Simon, 19721 M. Reed and B. Simon. Methods of modern mathemetical physics I, Academic Press, 1972. [Roberts and Roepstorff, 19691 J . E. Roberts and G. Roepstorff. Some basic concepts of algebraic quantum theory, Commun. math. Phys. 11, 321-338, 1969. [Sharrna and Almeida, 1990al C. S. Sharma and D. F. Almeida. Additive isonzetries o n a quaternionic Hilbert space, J. Math. Phys. 31 ( 5 ) 1035-1041, 1990. [ ~ h a r m and a Almeida, 1990bl C. S. Sharma and D. F. Almeida. A direct proof of Wigner's theorem o n maps which preserve transition probabilities between pure states of quantum systems, Ann. Phys. 197, 300-309, 1990. [Simon, 19761 B. Simon. Quantum dynamics: from autonaorphism to hamiltonian i n Studies i n Mathematical Physics. Essays i n Honor of Valentine Bargmann, eds E. H. Lieb, B. Simon, A. S. Wightman, Princeton Series i n Physics, Princeton University Press, Princeton, 327-349, 1976. [Schaefer,19641 H . Schaefer. Topological vector spaces, T h e Macmillan Company, 1964. [Semrl, 20031 P. Semrl. Generalized symmetry transfomaations on quaternionic indefinite inner product spaces: A n extension of quaternionic version of Wigner's theorem, Comrn. Math. Phys. 242, 579-584, 2003. [Uhlhorn, 19621 U . Uhlhorn. Representation of symmetry tmnsfornzations i n quantum mechanics, Arkiv for Fysik 23 (30), 307-340, 1962. [van der Broek, 1984a] P. M. van der Broek. Symmetry transformations in indefinite metric spaces: A generalization of Wigner's theorem, Physica A 127, 599-612, 1984. [van der Broek, 1984b] P. M. van der Broek. Group representations in indefininite metric spaces, J . Math. Phys. 25 ( 5 ) , 1205-1210, 1984. [Varadarajan, 19851 V . S. Varadarajan. Geometry of quantuna theory (2nd edition), SpringerVerlag, 1985. [Weinberg, 19951 S. Weinberg. The Quantum Theory of Fields, Cambridge University Press, 1995. [Wigner, 19311 E. Wigner. Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektren, Vieweg, Braunscheig, 1931. [wigner, 19391 E. Wigner. O n unitary representations of the inhomogeneow Lorentz Groups, Annals o f Math. 40 ( I ) , 149-204, 1939. [Wigner, 19591 E. Wigner. Group theory and its applications to quantum mechanics of atomic spectra, Academic Press Inc., New York, 1959. [Wigner, 19601 P. Wigner. Phenomenological distinction between unitary and antiunitary symmetry operators ,J. Math. Phys. 1, 414-416, 1960. [Wright, 19771 R. Wright. The structure of projection-valued states: A generalization of the Wigner's theorem, Int. Journ. Theor. Physics 16 ( a ) , 567-573, 1977. [Zhmud, 19921 E. M. Zhmud. O n a Theorem of Wigner, Selecta Mathernatica Sovietica 11 ( 2 ) , 71-82, 1992.

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HANDBOOK O F QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM STRUCTURES Edited by K. Engesser, D. M. Gabbay and D. Lehmann O 2007 Elsevier B.V. All rights reserved

477

PROPOSITIONAL SYSTEMS, HILBERT LATTICES AND GENERALIZED HILBERT SPACES Isar Stubbe and Bart Van Steirteghem

1 INTRODUCTION

Description of the problem. The definition of a Hilbert space H is all about a perfect marriage between linear algebra and topology: H is a vector space together with an inner product such that the norm associated to the inner product turns H into a complete metric space. As is well-known for any vector space, the onedimensional linear subspaces of H are the points of a projective geometry, the collinearity relation being coplanarity. In other words, the set L(H) of linear subspaces, ordered by inclusion, forms a secalled projective lattice. Using the metric topology on H we can distinguish, amongst all linear subspaces, the closed ones: we will note the set of these as C(H). In fact, the inner product on H induces an orthogonality operator on L(H) making it a Hilbert lattice, and the map ( )IL: L(H) +L(H): A I+ A" is a closure operator on L(H) whose fixpoints are precisely the elements of C ( H ) . In quantum logic it is the substructure C(H) L(H) - and not L(H) itself which plays an important rSle; it is called a propositional system [Piron, 19761. In this survey paper we wish to explain the lattice theoretic axiomatization of such a propositional system: we study necessary and sufficient conditions for an ordered set (C, 5 ) to be isomorphic to (C(H), C) for some (real, complex, quaternionic or generalized) Hilbert space H . As the above presentation suggests, this matter is intertwined with some deep results on projective geometry.

Overview of contents. Section 2 of this paper presents the relevant definitions of, and some basic results on, abstract (also called 'modern' or 'synthetic') projective geometry. Following C1.-A. Faure and A. Frolicher's [2000]reference on the subject, we define a 'projective geometry' as a set together with a ternary collinearity relation (satisfying suitable axioms). The one-dimensional subspaces of a vector space are an example of such a projective geometry, with coplanarity as the ternary relation. After discovering some particular properties of the ordered set of 'subspaces' of such a projective geometry, we make an abstraction of this

Isar Stubbe and Bart Van Steirteghem

Figure 1. A diagrammatic summary ordered set and call it a 'projective lattice'. We then speak of 'morphisms' between projective geometries, resp. projective lattices, and show that the category ProjGeom of projective geometries and the category ProjLat of projective lattices are equivalent. Vector spaces and 'semilinear maps' form a third important category Vec, and there is a functor Vec--+ProjGeom. The bottom row in figure 1 summarizes this. A projective geometry for which every line contains at least three points, is said to be 'irreducible'. We deal with these in section 3, for this geometric fact has an important categorical significance [Faure and Frolicher, 20001: a projective geometry is irreducible precisely when it is not a non-trivial coproduct in ProjGeom, and every projective geometry is the coproduct of irreducible ones. By the categorical equivalence between ProjGeom and ProjLat, the "same" result holds for projective lattices. The projective geometries in the image of the functor Vec --+ ProjGeom are always irreducible. Having set the scene, we deal in section 4 with the linear representation of projective geometries (of dimension at least 2) and their morphisms, i.e. those objects and morphisms that lie in the image of the functor Vec-+ ProjGeom. The First Fundamental Theorem, which is by now part of mathematical folklore, says that precisely the 'arguesian' geometries (which include all geometries of dimension at least 3) are "linearizable". In finite dimension, it is also known as the VeblenYoung Theorem, after the fathers of the axioms of projective geometry [Veblen and Young, 1910-181. The Second Fundamental Theorem characterizes the "linearizable" morphisms. [Holland, 1995, 331 and [Faure, 20021 have some comments on the history of these results. We outline the proof of the First Fundamental Theorem as given in [Beutelspacher and Rosenbaum, 19981; for a short proof of the Second Fundamental Theorem we refer to [Faure, 20021. Again following [Faure and Frolicher, 20001, we turn in section 5 to projective geometries that come with a binary orthogonality relation which satisfies certain axioms: so-called 'Hilbert geometries'. The key example is given by the projective geometry of one-dimensional subspaces of a 'generalized Hilbert space' (a notion due to C. Piron [1976]), with the orthogonality induced by the inner product. The projective lattice of subspaces of such a Hilbert geometry inherits an orthogonality operator which satisfies some specific conditions, and this leads t o the notion of 'Hilbert lattice'. The elements of a Hilbert lattice that equal

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their biorthogonal are said to be '(biorthogonally) closed'; they form a 'propositional system' [Piron, 19761: a complete, atomistic, orthomodular lattice satisfying the covering law. Considering Hilbert geometries, Hilbert lattices and propositional systems together with suitable ('continuous') morphisms, we obtain a triple equivalence of the categories HilbGeorn, HilbLat and PropSys. And there is a category GenHilb of generalized Hilbert spaces and continuous semilinear maps, with a functor GenHilb+HilbGeom. Since a Hilbert geometry is a projective geometry with extra structure, and a continuous morphism between Hilbert geometries is a particular morphism between (underlying) projective geometries, there is a faithful functor HilbGeorn -+ ProjGeom. Similarly there are faithful functors HilbLat--+ ProjLat and GenHilb-Vec too, and the resulting (commutative) diagram of categories and functors is drawn in figure l. Then we show in section 6 that a Hilbert geometry is irreducible (as a projective geometry, i.e. each line contains at least three points) if and only if it is not a non-trivial coproduct in HilbGeorn; and each Hilbert geometry is the coproduct of irreducible ones. By categorical equivalence, the "same" is true for Hilbert lattices and propositional systems. In section 7 we present the Representation Theorem for propositional systems or, equivalently, Hilbert geometries (of dimension at least 2): the arguesian Hilbert geometries constitute the image of the functor GenHilb +HilbGeorn. For finite dimensional geometries this result is due to G. Birkhoff and J. von Neumann [I9361while the more general (infinite-dimensional) version goes back to C. Piron's [1964,1976]representation theorem: every irreducible propositional system of rank at least 4 is isomorphic to the lattice of closed subspaces of an essentially unique generalized Hilbert space. We provide an outline of the proof given in [Holland, 1995, 531. The final section 8 contains some comments and remarks on various points of interest that we did not address or develop in the text. Required lattice and category theory. Throughout this chapter we use quite a few notions and (mostly straightforward) facts from lattice theory. For completeness' sake we have added a short appendix in which we explain the words marked with a "t" in our text. The standard references on lattice theory are [Birkhoff, 1967; Gratzer, 19981, but [Maeda and Maeda, 1970; Kalmbach, 19831 have everything we need too. Finally, we also use some very basic category theory: we speak of an 'equivalence of categories', compute some 'coproducts', and talk about 'full' and 'faithful' functors. Other categorical notions that we need, are explained in the text. The classic [Mac Lane, 19711 or the first volume of [Borceux, 19941 contain all this (and much more). Acknowledgements. As students of the '98 generation in mathematics in Brussels, both authors prepared a diploma dissertation on topics related to operational quantum logic, supervised and surrounded by some of the field's most outstanding researchers - Dirk Aerts, Bob Coecke, Frank Valckenborgh. Moreover, the quantum physics group in Brussels being next of kin to Constantin Piron's group

Isar Stubbe and Bart Van Steirteghem

P

Figure 2. Illustration of (G3) in Geneva, we also had the chance to interact with the members of the latter Claude-Alain Faure, Constantin Piron, David Moore. It is with great pleasure that we dedicate this chapter to all those who made that period unforgettable. We thank Francis Buekenhout, Mathieu Dupont, Claude-Alain Faure, Chris Heunen and Frank Valckenborgh for their comments and suggestions.

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2 PROJECTIVE GEOMETRIES, PROJECTIVE LATTICES It is a well-known slogan in mathematics that "the lines of a vector space are the points of a projective geometry". To make this statement precise, we must introduce the abstract notion of a 'projective geometry'. DEFINITION 1. A projective geometry (G,1) is a set G of points together with a ternary collinearity relation 1 C G x G x G such that (Gl) for all a, b E G, l(a, b, a), (G2) for b1 ~1 q G7 if P, q)l ~1 q) and p # q, then b1 P)> (G3) for all a, b, c, d,p E G, if I(p, a, b) and l(p, c, d) then there exists a q E G such that l(q, a, c) and l(q, b, d). Often, since no confusion will arise, we shall speak of "a projective geometry G", without explicitly mentioning its collinearity relation I. The axioms for the collinearity relation - as well as many of the calculations further on - are best understood by means of a simple picture, in which one draws "dots" for the points of G, and a "line" through any three points a, b, c such that l(a, b, c ) . With this intuition (which will be made exact further on), (Gl) and (G2) say that two distinct points determine one and only one line, and (G3) is depicted in figure 2. EXAMPLE 2. Let V be a (left) vector space over a (not necessarily commutative) field K. The set of lines of V endowed with the coplanarity relation forms a projective geometry; it will be denoted further on as P(V).Note that the collinearity relation is trivial when dirn(V) 5 2.

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The example P(V) is very helpful for sharpening the intuition on abstract projective geometry. For example, it is clear that the collinearity relation in P(V) is symmetric; but in fact this property holds also in the general case. LEMMA 3. A ternary relation 1 on a set G satisfying (GI-2) is symmetric, meaning that for a l , az, as E G, if l(al, az, as) then also l(a,(l), a,(z), a+)) for any permutation a on {1,2,3).

Proof. The group of permutations on {1,2,3) being generated by its elements (123) I+ (132) and (123) H (312), we only need to check two cases. This is a simple exercise. H It is not hard to show that (G3) follows from (GI-2) when card{a, b, c, d,p) # 5 or when {a, b, c, d) contains three (different) points that belong to 1. For a projective geometry (G, l ) , any two distinct points a, b E G determine the projective line a * b := {x E G I l(s,a, b)). For notational convenience, we also put that a * a := {a). It is a useful corollary of Lemma 3 that for a, b, c E G, if a # c then a € b*cimplies b E a*c. Now we define a subspace S of G to be a subset S C G with the property that

Trivially, any projective geometry G has the empty subspace 0 C_ G and the total subspace G G. Moreover, all a*b G are subspaces; these include all singletons {a) = a k a . The set of all subspaces of G will be denoted L(G). Since subspaces of G are particular subsets of G, C(G) is ordered by inclusion. The following proposition collects some features of the ordered sett (C(G), G), but first we shall record a key lemma. LEMMA 4. In the lattict! C(G) of subspaces of a projective geometry G,

ni

i. for any family of subspaces (Si)iEl, Si is a subspace, ii. for a directed family of subspaces (Si)iEl, Si is a subspace, iii. for two non-empty subspaces S and T, U{a * b I a E S,b E T) is a subspace.

Ui

Proof. The prooh of the first two statements are straightforward. As for the third statement, we must prove that, if l(x, a l l bl), l(y, az, b2) and l(z,x, y), with a l , a2 E S and bl, b2 E T, then l(z, as, b3) for some a3 E S and b3 E T . The picture in figure 3 suggests how to do this, using the symmetry of the collinearity relation H and applying (G3) over and over again. It follows that for a subset A 5 G of a projective geometry G

Isar Stubbe and Bart Van Steirteghem

Figure 3. Illustration of the proof of Lemma 4 is the smallest subspace of G that contains A: it is its secalled projective closure1.The third statement in Lemma 4 is often referred to as the projective law. In terms of the projective closure it may be stated as: for non-empty subspaces S and T of G, cl(SUT) = U { a * b ~ a € ~ , b € ~ } . PROPOSITION 5. For any projective geometry G, (L(G), c) is a completet, atomistic!, continuoud, modula4 lattice.

Proof. The order on L(G) is complete, because the intersection of subspaces is their infimum+;thus the supremumtof a family (Si)iE1 E L(G) is Vi Si = cl(Ui Si). This makes it at once clear that any subspace S E L(G) is the supremum of its points: S = cl(S) = VaEs{a}; and singleton subspaces being exactly the atoms* of L(G) this also shows that L(G) is atomistic. The continuity of L(G) follows trivially from the fact that directed suprema in L(G) are simply unions. Finally, to show that L(G) is modular, it suffices to verify that for non-empty subspaces S , T , U C G, if S C T then ( S V U ) n T C S v ( U n T ) . We aregoing to use the projective law a couple of times. Suppose that x E (S V U) n T; so x E T, but also x€SVU,whichmeansthatx€a*bforsomea€Sandb€U. Ifx=athen x~S~S~(U~T);ifx#athenx€a*bimpliesthatb€a*xCSVT=T (using that S T) so that x E a * b S V (U n T) in this case too. DEFINITION 6. An ordered set (L, 5 ) is a projective lattice if it is a complete, atomistic, continuous, modular lattice. There are equivalent formulations for the definition of 'projective lattice'; we shall encounter some further on in this section. Here we shall already give one alternative for the continuity condition, which is sometimes easier to handle and will be used in the proofs of Theorem 16 and Lemma 24. lThis terminology is well-chosen, for the mapping A H cl(A) does indeed define a closure operatort on the set of subsets of G; see also Comment 89.

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LEMMA 7. A complete atomistic lattice L is continuous if and only if its atoms are compact, i.e. if a is an atom and (xi)iEl is a directed family in L, then a 5 Vi xi implies a 5 xk for some k E I. Proof. If L is continuous and (with notations as in the statement of the lemma) a< - V ixi, then a = a A (Vixi) = V i ( aA xi), so there must be a k E I for which a A xk # 0, whence a 5 xk (for a is an atom). Conversely, Vi(Y A xi) 5 y A (Vixi) holds for any element y E L. Suppose that this inequality is strict. By atomisticity x j a) 5 yA(Vi xi). This of L there must exist an atom a such that a $ V i ( ~ ~ and implies in particular that a 5 y and a 5 Vi xi, and by hypothesis a is compact so that a 5 xk for some k E I. But then a 5 y A xk 5 V i (A~xi) is a contradiction. Thus necessarily V i ( yA xi) = y A (Vixi) for every y E L.

EXAMPLE 8. For a K-vector space V , the set L ( V ) of linear subspaces, ordered by set-inclusion, is isomorphic to the projective lattice L ( P ( V ) )of subspaces of the projective geometry P ( V ) of Example 2: the mappings

are well-defined, preserve order*and are each other's inverse. With slight abuse of notation we shall write L ( V ) even when we actually mean L ( P ( V ) ) . So Proposition 5 states that a projective geometry G determines a projective lattice L(G); but the converse is also true. First we prove a lattice theoretical lemma that exhibits the strength of the modularity condition.

LEMMA 9. Let L be a complete atomistic lattice, and B(L) its set of atoms. i. If L is modular, then it is both upper semimodula~and lower semimodula~. ii. L is upper semimodular if and only if it satisfies the covering lad. iii. If L is lower semimodular and satisfies the covering law then it has the intersection property: for any x E L and p, q E G(L) with p # q, if p I q V x then(pVq)Ax#O. iv. If L has the intersection property, then for a, b, c E G(L) with a # b, if a 5 b V c then also c 5 aV b. v. If L has the intersection property, then G(L)forms a projective geometry for the ternary relation (1) l(a,b, c) if and only if a 5 b V c or b = c.

Proof. (i) For any lattice L, the maps p: [uA v ,v]-+ [u,uV v]:x

t+

x V u and $: [u,uV v]--t [uA v,v]:y

I+

yAv

are well-defined and preserve order. If L is modular then moreover $ ( p ( x ) )= ( x V u )Av = X V ( u A x )= x; similarly p($(y)) = y. So the two segments are isomorphic

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lattices. Now clearly u A v < v @ card[u~v,v]= 2 @ card[u,uVv] = 2 @ u3 +Arg is essentially surjective and essentially injective on objects, full, and faithful up to scalar multiple on semilinear maps whose image is not a line. 5 HILBERT GEOMETRIES, HILBERT LATTICES, PROPOSITIONAL SYSTEMS

A (real, complex, quaternionic or generalized) Hilbert space H is in particular a vector space, so by Example 8 its one-dimensional linear subspaces form a projective geometry P(H). But the orthogonality relation on the elements of H , defined as x I y if and only if the inner product of x and y is zero, obviously induces an orthogonality relation on P(H): A I B in P ( H ) when a I b for some a E A \ (0) and b E B \ (0). We make an abstraction of this. DEFINITION 51. Given a binary relation I G x G on a projective geometry G and a subset A c G, we put A'- := {b E G I Va E A : b I a). A Hilbert geometry G is a projective geometry together with an orthogonality relation Ic G x G such that, for all a, b, c,p E G, (01) (02) (03) (04) (05)

if if if if if

a I b then a # b, a I b then b Ia, a # b, a I p, b I p and l(a, b, c) then c I p, a # b then there is a q E G such that l(q, a, b) and q I a, S 5 G is a subspace such that SL'-= S, then S V SL= G.

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Very often we shall simply speak of a "Hilbert geometry G", leaving both the collinearity I and the orthogonality Iunderstood. A subspace S C G is said to be (biorthogonally) closed if SLL= S. Axioms (01-4) in the above definition say in particular that a Hilbert geometry is a 'state space' in the sense of [Moore, 19951 as we explain in Comment 90. The fifth axiom could have been written as: S = SLLif and only if SvSL = G, because (as we shall show in Lemma 56 (iv) in a more abstract setting) the necessity is always true. We make some more comments on these axioms in section 8. The term 'Hilbert geometry' is well-chosen, as C. Piron's now famous example 11964, 19761 shows. DEFINITION 52. A generalized Hilbert space (also called orthomodular space) (H, K, *, ( , )) is a vector space H over a field K together with an involutive anti-automorphism K +K : a H a* and an orthomodular Hermitian form H x H --+ K: (x, y) ct (x, y), that is, a form satisfying (S1) ( X x l + x 2 , ~ ) = X ( x l , ~ ) + ( x 2forallx1,22,y ,~) EH,XEK, (S2) (y, x) = (x, y)* for all x, y E H, and such that, when putting SL := {x E H I Vy E S : (x, y) = 0) for a linear subspace S C H , (S3) S = SLLimplies S @ SL= H. Note that an orthomodular Hermitian form is automatically anisotropic, (S4) (x, x)

# 0 for all x E H \ {0),

and that in the finite dimensional case the converse is true too. I. Amemiya and H. Araki 119661 proved that when K is one of the "classical" fields equipped with its "classical" involution (W with identity, (I3 and W with their respective conjugations), the definition of 'generalized Hilbert space' is equivalent to the "classical" definition of a Hilbert space as inner-product space which is complete for the metric induced by the norm. While H. Keller [I9801 was the first to construct a 'honc1assica1~'generalized Hilbert space, M. Sol& [I9951 proved that an infinite dimensional generalized Hiibert space H is "classical" precisely when H contains an orthonormal sequence. We refer to [Holland, 19951 for a nice survey, and to A. Prestel's [2006] contribution to this handbook for a complete proof of SolGr's theorem. For a comment on the lattice-theoretic meaning of Sol&r'stheorem, see Comment 98. EXAMPLE 53. For a generalized Hilbert space H, the projective geometry P ( H ) together with the obvious orthogonality relation forms a Hiibert geometry: axioms (01-3) are immediate, (04) follows from a standard Gram-Schmidt trick and (05) is also immediate since S V SL= S @ SLfor any linear subspace S of H. From our work in section 2 we know that, since a Hilbert geometry G is in particular a projective geometry, the lattice of subspaces L(G) is a projective

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lattice. Because of the orthogonality relation on G , there is some extra structure on L ( G ) ; the following proposition identifies it. PROPOSITION 54. If G is a Hilbert geometry with orthogonality relation I , then the operator I : C ( G )+C ( G ): S I+ sLsatisfies, for all S, T E L ( G ) ,

i. ii. iii. iv. v.

c

S SLL, i f S G T then T'- G S L , S n sL= 0, if S = SSI and a E G then { a ) V S = ( { a ) V S ) I L . if S = SLL then S v SL = G.

Proof. All is straightforward, except for (iv). We need to prove that ( { ~ ) v S C) ~ ~ { a ) V S for S = SL'-. If a E S then this is trivial so we suppose from now on that a @ S = S L L , i.e. there exists p E S L such that a , lp. Let b E ( { a ) V S)"-; if b = a or b E S then obviously b E { a ) V S. If b # a and b $! S then we claim that (a* b) n is a singleton and moreover that its single element, call it q, belongs to S. This then proves the assertion, for q I p implies q # a, which makes q E a*b imply that b € a * q G { a ) V S . is non-empty4, because in case that a , lp , lb we can always Now (a* b) n pick x E p*a and y E p* b such that x , y E {p)'- by ( 0 4 ) ;then x # a and y # b so p E ( a * x ) n ( b * y ) and ( G 3 ) thus gives a q ( a~* b ) n ( x * y ) C ( a * b ) n f p l L (for { p ) l is a subspace by ( 0 3 ) ) . Would ql # 92 E ( a * b) n { p I L , then l(q1,q2, a ) by ( G 2 ) hence a E a contradiction. So we conclude that ( a * b) n = {q). We shall show that q E S = S L L , i.e. for any r E S L we have q I r. For r = p this is true by construction; for r # p we may determine, by the "same" argument as above, a (unique) point s E { a l l n ( p * r ) C ({a)'- n s'-) = ( { a ) V s ) ~ .The latter equality can be shown with a simple calculation, but we also give a more abstract proof in Lemma 56 (iii). Because a, b E ( { a )V S ) I L it follows that a Is and b I s; hence we get q I s from q E a * b and ( 0 3 ) . But also s # p follows, thus s E p * r implies r E p * s , and because we know that q I p too, we finally obtain q Ir , again from ( 0 3 ) . This proposition calls for a new definition. DEFINITION 55. A projective lattice L is a Hilbert lattice if it comes with an orthogonality operator '- :L --+ L: x I+ xL satisfying, for all x, y E L,

(HI) (H2) (H3) (H4) (H5)

x 5 xLL, if x 5 y then y L 5 xL, x A x L = 0, if x = x L L and a is an atom of L then a V x = ( a V x ) l L , if x = xu then x v xL = 1.

4What we really prove here is that for a, b,p with a # b in a Hilbert geometry G there always exists some q E a * b such that q Ip. This statement, which is obviously stronger than (04), is often used instead of (04). See Comment 91 for a relevant comment.

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Usually we shall simply speak of "a Hilbert lattice L", and leave the orthogonality operator understood. The crux of Proposition 54 is thus that the projective lattice of subspaces of a Hilbert geometry is a Hilbert lattice. Having Proposition 10 in mind, it should not come as a surprise that there is a converse to this. However, we shall not give a direct proof of such a statement, for we wish to involve yet another mathematical structure. Again the source of inspiration is the concrete example of Hilbert spaces: the subspaces S C H for which S = S1l are particularly important, for they are precisely the subspaces which are closed for the norm topology on H (see [Schwartz, 1970, p. 3921 for example). Also in the abstract case they are worth a closer look. By a (biorthogonally) closed element of a Hilbert lattice (L, I ) we shall of course mean an x E L for which x = xLL. We write C(L) C L for the ordered set of closed elements, with order inherited from L. We shall now discuss some features of this ordered set that - as it will turn out - describe it completely. First we prove a technical lemma. LEMMA 56. For a Hilbert lattice L,

i. if x E L then x1 E C(L), ii. oLL = 0, oL = 1 and l1 = 0, iii- (Vi x i ) l = X: for (xi)iE1E L, iv. if x v xL = 1 then x = xL1, v. the map I L : L -+L: x H xL1 i s a closure operator with jixpoints C(L). Proof. Statements (i) and (ii) are almost trivial. For (iii) one uses (HI-2) over and again to verify 2 and 5 as follows:

As for (iv), the assumption together with (H3) and modularity in L (for x 5 xL1) give x = x v 0 = x v (xL A xLL) = (x v xl) A xLL = 1A xLL = xLL. Finally, it straightforwardly follows from (H2) that ~ : L + C ( L ) : X H X ~and ~ $:C(L)+L:y+y are maps that preserve order, and they satisfy p(x) 5 y H x 5 $(y) for any x E L and y E C(L). So these maps are adjoint, cp -I $, and since moreover p is surjective and $ injective, the composition $ op: L --+L: x H xL1 is a closure operator with fixpoints C(L), as claimed in (v). For closed elements (xi)iEr E L we shall write Vixi for (Vi xi)ll, and in particular x ~ for y (x t:y)LL. PROPOSITION 57. For any Hilbert lattice L, the ordered set (C(L), 5 ) together with the restricted operator I:C(L) +C(L): x H xL is a complete, atomistic, orthomodula$ lattice satisfying the covering law.

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Proof. By (v) of 56, C(L) is a complete lattice inheriting infima from L, and with suprema given by v . Moreover, (H2-3) assert that x I+ x L is an orthocomplementationt on C(L). It is straightforward from (H4) and Lemma 56 (ii) that the atoms of L are closed; and conversely is it clear that the atoms of C(L) are atoms of L too. So C(L) is atomistic, because L is. In the same way, since L has the covering law (cf. Lemma 9) and the atoms of L are precisely those of C(L), again (H4) assures that C(L) has the covering law too. Finally, if x 5 y in C(L) then by the modular law in L and (H5)

i.e. the orthomodular law holds in C(L).

rn

EXAMPLE 58. By (H4) it follows that, if a l , ..., a, are atoms of a Hilbert lattice L, then (each one of them is closed and) a l V . ..Van = a1 V ...V a,. If L is a Hilbert lattice of finite rank, then every x E L can be written as a finite supremum of atoms (this is true for any atomistic lattice satisfying the covering law of finite rank, see e.g. [Maeda and Maeda, 1970, section 8]), hence x = x L L ; i.e. L E C(L). So if G is a Hilbert geometry of finite dimension, then every subspace of G is biorthogonally closed; in particular is this true for P ( H ) when H is a (generalized) Hilbert space of finite dimension. Inspired by the result in Proposition 57 we now give another definition due to C. Piron [1964, 19761. DEFINITION 59. An ordered set (C, 5 ) with an operator '-: C-C: x o x'- is a propositional system if it is a complete, atomistic, orthomodular lattice that satisfies the covering law (with x o x L as orthocomplementation). We shall speak of "a propositional system C", always using x L as notation for the orthocomplement of x E C. And we shall continue to write V i x i for the supremum in C, and Ai xi for the infimum. EXAMPLE 60. The closed subspaces of a generalized Hilbert space H form a propositional system, that we shall write as C(H) instead of C(L(H)). According to Proposition 57 and Definition 59, the closed elements of a Hilbert lattice form a propositional system. Earlier we proved (cf. Proposition 54 and Definition 55) that the subspaces of a Hilbert geometry form a Hilbert lattice. It is now time to come full circle: we want to associate a Hilbert geometry to a given propositional system. The lattice-theoretical results in Lemma 9 will be useful here too. PROPOSITION 61. The set G(C) of atoms of a propositional system C fomn a Hilbert geometry for collinearity and orthogonality given by

Proof. By definition C is complete, atomistic and satisfies the covering law; therefore it is upper semimodular by (ii) of Lemma 9. But C is also orthocomplementedt

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, so it is isomorphic to its opposite+(by C --+COP: x I+ xL): upper semimodularity thus implies lower semimodularity. Then C must have the intersection property by (iii) of Lemma 9, and so its atoms form a projective geometry for the indicated collinearity. Now we must check the axioms for the orthogonality relation; the first three are (almost) trivial. For (04), if a # b in B(C) then b 5 avaL = 1 hence, by the intersection property, ( a ~ b A) a'- # 0; the atomisticity of C gives us a q E B(C) such that q 5 avb and q 5 a'-, as wanted. Finally, (05) requires some more sophisticated calculations. First note that, for any subspace S C B(C) and element a E g(C), Thus we always have that S'- = {a E B(C) I a 5 ( V S ) ~ ) ,which by atomisticity of C means that V (SL) = (VS)'-; in particular is S closed, S = SLL,if and only if S = {a E B(C) I a 5 VS). If S is a trivial subspace, then it is clear that S V S'- = B(C); so from now on, let S = SL'- be non-trivial. By the projective law, valid in L(B(C)) as in any other projective lattice, S V SL= B(C) just means that for any p E B(C) there exist a, b E B(C) such that a 5 VS, b 5 (VS)'and p 5 avb. And this is indeed true in the propositional system C; to simpllfy notations we shall write x := V S in the argument that follows5. Suppose first that x A (xLvp) $ x'-, then (by C's atomisticity) there must exist an a E B(C) such that a 5 x r\ (xLvp) and a xL. If a = p then p 5 x and we can pick any avb as wanted. If a # p then from a 5 xLvp atom b 5 x'- to show that p and the intersection property we get an atom b 5 xL A ( a ~ p ) but ; certainly is a # b (because a 5 x and b 5 xL) so b 5 aVp is equivalent to p 5 avb by Lemma 9 (iv), as wanted. Next suppose that x A (xLvp) 5 x'-; this means that xL = x L v (x A (xLvp)) = xLvp by orthomodularity in the second equality, so Picking any atom a 5 x and putting b := p we have p 5 avb as wanted. p 5 XI.




it is monotone and satisfies cl(cl(A)) c cl(A) A, a E cl(A) implies a E cl(B) for some finite subset B c A, x # cl (A) and x E cl (A U {b)) imply b E cl (A u {x)), cl(0) = 0 and cl({a)) = { a ) for all a E G, for non-empty A, B c G, cl(A U B) = U{cl({a, b)) I a E cl(A), b E cl(B)).

A set G together with an operation cl: 2G+2G satisfying (i) is called a closure space; if on top of that it satisfies (ii-iii) then it is a matroid. A closure space that also satisfies (iv) is a simple closure space; and a simple matroid is often called a geometry. If A I+ cl(A) satisfies the whole lot (i-v) then it is a projective closure, and one can prove that any projective closure space (G, cl) is necessarily provided by a projective geometry. That is to say, there is an equivalence of categories ProjGeorn ProjClos of projective geometries on the one hand and projective closure spaces on the other (with appropriate morphisms). But also the 'weaker' structures (matroids, geometries) are interesting in their own right; in particular can a whole deal of "dimension theory" for projective geometries (cf. Definition 28) be carried out for structures as basic as matroids. This is the

-

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subject of C1.-A. Faure and A. Frolicher's [1996], see also their [2000, chapters 3 and 41. Comment 90. State spaces and property lattices. In Definition 51 of Hibert geometry, it follows from (01-4) that a Hilbert geometry is a state space in the sense of [Moore, 19951: if a # b then l(q, a, b) and q I a for some q E G by (04), but would q Ib as well then q I q by (02-3) (and the symmetry of 2) which is excluded by (01). That is to say, the relation I is irreflexive, symmetric and separating (in the sense that a # b implies the existence of some q such that a I q ,k! b). Moore [I9951 proves that the biorthogonally closed subspaces of a state space form a so-called property lattice: a complete, atomistic and o r t h e complemented lattice. Of course, a propositional system (cf. Definition 59) is a particular example of such a 'property lattice'. More precisely, state spaces and property lattices are the objects of equivalent categories State and Prop of which the equivalence of HilbGeom with PropSys is a restriction. For the relevance of State and Prop in theoretical physics see [Moore, 19991. Comment 91. Fewer axioms for geometries with orthogonality. F. Buekenhout [I9931 explains how A. Parmentier and he showed that, remarkably, (G3) of Definition 1 is automatically true for a set G with a collinearity 1 satisfying just (GI-2) and an orthogonality I satisfying (cf. Definition 51)

(02) (03) (06) (07)

if a Ib then b I a, ifa#b, a l p , b I p a n d c ~ a * b t h e n c I p , i f a , b , p ~ Ga n d a # b t h e n t h e r e i s a q ~ a * b w i t h q I p , for all a E G there is a b E G with a ,k! b.

Clearly, (01) implies (07), and in the proof of Proposition 54 we have shown that (06) too is valid in any Hilbert geometry. Comment 92. Geometries "with extra structure". A Hilbert geometry is, by Definition 51, a projective geometry with extra structure - a lot of extra structure, actually. There are many notions of 'projective geometry with extra structure' that are weaker than Hilbert geometries but still have many interesting properties. A large part of [Faure and Frolicher, 20001 is devoted to the study of such things as Mackey geometries, regular Mackey geometries, orthogeometries and pure orthogeometries: structures that lie between projective geometries and Hilbert geometries. Several of these 'geometries with extra structure' can be represented by appropriate 'vector spaces with extra structure.' In that spirit, [Holland, 1995, 3.61 and [Faure and Frolicher, 2000, 14.1.81 are slight generalizations of Piron's representation theorem (cf. Theorems 81 and 82) which include skew-symmetric forms. Comment 93. On orthomodularity. Given a vector space V with an anisotropic Hermitian form (i.e. a form satisfying (Sl, S2 and S4) in Definition 52) and induced orthogonality I we have that (P(V), I)satisfies (05) if and only if L(V) satisfies (H5) if and only if the lattice of closed subspace C(V) is orthomodular if and only if the Hermitian form satisfies (S3). In other words, and this is a

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key insight of Piron's [1964], orthomodularity of C(V)is what distinguishes the generalized Hilbert spaces among the (anisotropic) Hermitian spaces. This is one of the reasons why orthomodular lattices have been heavily studied; the standard reference on the subject is [Kalmbach, 19831.

Comment 94. Projectors. For a projective geometry G together with a binary relation I on G that satisfies (01-4) in Definition 51, a (necessarily closed, cf. Lemma 56) subspace S G satisfies SvSL = G if and only if for every a E G \ S L the subspace ({a)vSL)nS is non-empty. In this case, ({a)vSL)nS is a singleton, ) its single element gives a partial map and writing ~ ( a for T:G - + S:a

I+

T (a)

with kernel s1

which is a retract to the inclusion i: S--4 G.

Proof. Suppose that S V SL = G and that a E G \ (SU S1) (if a E S then all is trivial). By the projective law, a E x * y for some x E S and y E SL,whence x E a * y C {a) V SL, so x E ({a) V SL) n S # 0. Conversely, suppose that a E G \ ( S u S L ) . Pick any x E ({a)vS1)nS: thus x E S a n d x E a*y for some y E SL,whence a E x * y C S V SL (using the projective law twice), which proves that S VS1 = G. Would ~ 1 ~ be x 2different elements of ({a) V SL) n S for some a $ SL,then X I , xz E S and there exist yl, yz E {a} V SL such that xi E a * yi for i = 1,2. Because a # SL,a is necessarily different from the yi's. But a is also different from the xi's: if a = XI for example, then xz E XI * yz from which y2 E x l * x2 S, which is impossible since S n SL = 0 by (01). So we can equivalently write that a E (XI * yl) n (x2 * y2); and by axiom (G3) of Definition 1 we get a point b E (XI * x2) n (yl * yz). But such b lies in both S and SL, which is impossible. Hence the non-empty set ({a) V SL) n S is a singleton. In particular does this argument imply that {a) = ({a) V SL) n S if a E S: so the partial map T:G - + S sending a $ SLto the single element of ({a) V SL) n S is a retract to the inclusion i:S+G. Now i: S + G is a morphism of projective geometries when we let S inherit the collinearity from G, but in fact so is T:G- +S. Therefore pr := i o T:G- + G is an idempotent morphism of projective geometries with kernel SL and image S . It is moreover true that pr(a) I b H a I pr(b) for a, b # S1 (the morphism is "self-adjoint"), and so we have every reason to speak of the projector with image S and kernel SL.Much more on this can be found in [Faure and Frolicher, 2000, section 14.41.

Comment 95. More on projectors. Interestingly, there is a lattice-theoretic analog of Comment 94: A complete orthocomplemented lattice C is orthomodular if and only if for each x E C the map cp:, C+C: y H x A (xl V y) has a right adjoint, which then is the map $,: C +C: y I+ xL V (x A y). If C is moreover atomistic and satisfies the covering law, then cp, is a morphism of propositional systems.

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Proof. Clearly the maps cp, and $, preserve order. Now let C be a complete orthocomplemented orthomodular lattice, then

( ( y ) ) = x L V xA x A ( x L V y))) = x L V ( x h ( x L V y ) ) -xLVY > Y

(

(

where orthomodularity was used in (*). Similarly one shows cp, ($,(y)) 5 y, so we Conversely, if x 5 y in a complete orthocomplemented get the adjunction cp, -I lattice C, then using this information in (**) gives

+,.

Assuming that c p , ~ 4 $,l we get y 5 +,l(y) from the first line, thus y = $,l(y) if we combine it with the second line, which is the orthomodular law. Next suppose that C is a propositional system, let a E C be an atom and x E C. If a 5 xL then cp,(a) = 0. If a $ xL then a A xL = 0 so xL G a V xL. By lower semimodularity of C (see Proposition 61 and use [Faure and Frolicher, 2000, 1.5.71) it follows that either x A xL = x A (a V xL) or x A xL x A (a V xL); in any case we have shown that cp,(a) is 0 or covers 0. H A map like the cp,: C-+C in the statement above, is called a Sasaki projector, and its right adjoint is a Sasaki hook. These maps were introduced by U. Sasaki [1954], and extensively used in [Piron, 1976, 4-11 to latticetheoretically describe the effect of an "ideal measurement of the first kind" on a (quantum) physical system. See also [Coeckeand Smets, 20041 for a discussion of the (quantum logical) meaning of the adjunction of Sasaki projection and Sasaki hook. Comment 96. Another irreducibility criterion. A bounded lattice is, by definition, a lattice with a smallest element 0 and a greatest element 1. If L1 and Lz are bounded lattices then, with componentwise lattice structure, the cartesian product L1 x Lz is a bounded lattice too. An element z E L of a bounded lattice is central if there exist bounded lattices L1, La and an isomorphism (i.e. a bijection L such that z = cp(1,O). The set that preserves and reflects order) cp: L1 x La Z(L) of central elements, called the center of L, is an ordered subset of L that contains at least 0 and 1. A wealth of information on this topic can be found in [Maeda and Maeda, 1970, sections 4 and 51 or any other standard reference on lattice theory. One can easily figure out that the cartesian product L1 x L2 of bounded lattices is a projective lattice if and only if L1 and L2 are projective lattices (just view such an Li as a segment in L); and L1 x Lz is then a coproduct in ProjLat (see also Lemma 24). Hence L is an irreducible projective lattice if and only if Z(L) = {0,1) ("L has a trivial center"). One can moreover show that Z(L) forms a complete atomistic Boolean (i.e. complemented and distributive) sublattice of L; and the segments [0,a] L, with a an atom of Z(L), are precisely the 'maximal irreducible segments' of L (a notion that we did not bother defining in section 3); so L is the

Isar Stubbe and Bart Van Steirteghem

Figure 8. Categorical definition of 'semilinear map' coproduct in ProjLat of these segments. Details are in [Maeda and Maeda, 1970, 16.61 for example, where the term 'modular matroid lattice' is used synonymously for 'projective lattice'. For a propositional system C , one can work along the same lines to prove that C is irreducible if and only if 2 ( C ) = {0,1}; the center Z(C) is again always a complete atomistic Boolean sublattice of C; and C is the coproduct in PropSys of the segments [O,a] with a an atom of Z(C). C. Piron [1976, p. 291 has called the atoms of 2 ( C ) the superselection rules of the propositional system C. In geometric terms, a subspace S G of a projective geometry is a central element in C(G) if and only if also the set-complement SC:= G \ S is a subspace of G. And if G is a Hilbert geometry then S is central if and only if SC= SL,in which case S is necessarily a closed subspace. So Z(C(G)) Z(C(L(G))) for a Hilbert geometry G, proving at once that the center of a Hilbert lattice L is the same Boolean algebra as the center of the propositional system C(L) of closed elements in L. Comment 97. Modules on a ring. Vector spaces on fields are very particular examples of modules on rings; and modules on rings are very "categorical" objects: consider a (not necessarily commutative) ring R as a oneobject Ab-enriched category R, then a (left) module (M, R) is an Ab-presheaf M : R+Ab. (As usual, Ab denotes the category of abelian groups.) In the same vein, also semilinear maps between vector spaces are instances of an intrinsically categorical notion: viewing ring-modules (M, R) and (N, S ) as Ab-presheaves M : R+ Ab and N :S+Ab, a "semilinear map" (f,a ) : (M, R) -+ (N, S) ought to be defined as an Ab-functor a:R+S together with and Ab-natural transformation f : M =sN o a, cf. figure 8. It is thus natural to investigate whether and how one can associate a (suitably adapted notion of) 'projective geometry' to a general module, and a morphism of projective geometries to a semilinear map as defined above. M. Greferath and S. Schmidt's Appendix E in [Gratzer, 19981 and Faure's [2004] deal with aspects of this; in our opinion it would be enlightening to both algebraists and geometers to study the (Ab-enriched) categorical side of this. Comment 98. Lattice-theoretic equivalents to SolBr's condition. Recall that M. P. Sol& [I9951 proved that an infinite dimensional generalized Hilbert space is a "classical" Hilbert space (over R, cC or W) exactly when it has an or-

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thonormal sequence (see [Holland, 19951 for much more on this theorem and [Prestel, 20061 in this volume for a complete proof). As SolGr pointed out in the same paper, the "angle bisecting" axiom of R.P. Morash [I9731 provides an equivalent but lattice-theoretic condition. S. Holland [1995, $41 used "harmonic conjugates" to formulate another lattice-theoretic alternative. He also proposed [Holland, 1995, $51 a (non lattice-theoretic) "ample unitary group axiom": an infinite dimensional orthomodular space H over K is a L'classical" Hilbert space if and only if for any two orthogonal nonzero vectors a, b E H there exists a unitary map U : H -+H (see Definition 87) such that U(Ka) = Kb. R. Mayet [I9981 has proved the following lattice-theoretic alternative: an orthomodular space H is an infinite dimensional Hilbert space over W,C or W if and only if there exist a, b E C(H) where dim b 2 2 and an ortho-isomorphism f : C(H) +C(H) such that f ([o,bl is the identical map and f (a) 5 a. The condition on f l[O,bl guarantees that the semi-unitary map inducing f (by Wigner's theorem, see Theorem 88) is unitary. Similar characterizations using "symmetries" of the lattice C(H) were proposed in [Aerts and Van Steirteghem, 20001 and in [Engesser and Gabbay, 20021. The question whether the transitivity of the whole group of ortho-isomorphisms of C(H) still characterizes the "classical" Hilbert spaces among the infinite dimensional orthomodular spaces seems to be unanswered. 9 APPENDIX: NOTIONS FROM LATTICE THEORY Mostly to 6x terminology, we recall the notions from lattice theory we have used in this chapter; in the previous sections these are marked with a "t" when they are used for the first time. A partially ordered set, also called simply ordered set or poset, is a set P together with a binary relation 5 which is reflexive, antisymmetric and transitive. We also use the standard notation x < y for x 5 y and x # y. The opposite ordered set P O P has the same elements as P but with its order relation 4 reversed: forx,yEPwehavex5ye~y=$x. For a subset X of P we say that p E P is an upper bound of X if x 5 p for every x E X , we say that p is a least upper bound, or a supremum, or a join, of X if for every other upper bound q we have p 5 q. By antisymmetry a least upper bound is unique if it exists. The concept of greatest lower bound (also called infimum or meet) is defined dually. If the supremum of X exists and lies in X we call it the maximum of X , denoted by maxX. Dually, we can define min X , the minimum of X. A lattice is a poset L any two of whme elements x, y E L have a meet denoted by x A y and a join denoted by x V y. It is complete if any subset X C L has a join, then denoted by V X I and a meet A X . (In fact, if all joins exist in an ordered set L then so do all meets, and vice versa; thus an ordered set L is a complete lattice if and only if it has all joins, if and only if it has all meets.) Putting X = L we see that a complete lattice has a b o t t o m element 0 and a top element 1, that is, elements satisfying 0 x 5 1 for every x E L. For a 5 b in a lattice L,

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the interval or segment [a,b] is the lattice {x E L I a 5 x 5 b } . A map f : PI -+P2 between two ordered sets is said to preserve order, or is called monotone, if for any x, y E PI, x 5 y implies f (x) 5 f (y). It is an isomorphism of ordered sets (or of lattices when appropriate) if it moreover has an order-preserving inverse. For two elements x, y of P we say that y covers x and we write X Gy when x < y but never x < p < y for p E P. If P is a poset with bottom element 0,we call a E P an a t o m if a covers 0. If P is a poset with top element 1then c is a coatom if 1 covers c. A lattice L with bottom element 0 is called6 atomistic if every element x E L is the join of the atoms it contains: x = V{a E L I a atom, a 5 x). A nonempty subset D C P of a poset is called directed if for any x, y E D, there exists z E D such that x 5 z and y 5 z. A complete lattice L is called continuous (some say meet-continuous) if for any directed set D L and any a~Lwehavea~(VD)=V{a~dId~D}. A lattice L is called modular if, for every x, y, z E L, x

5 z implies xV (y Az) = (xV y) A z.

The following are weaker notions: L is upper semimodular if u A v Q v implies u G u V v; and it is lower semimodular if u Q u A v implies u A v G v. A lattice L with 0 satisfies the covering law if for any x E L and any atom a E L we have a Ax = 0 implies x G a V x .

An orthocomplementation on a lattice L with 0 and 1is a map L + xL which satisfies, for all x, y E L,

L: x

H

i. x 5 y implies yL 5 xL, ii. ( x L ) l = x, ii. x v x L = l a n d x ~ x ~ = 0 .

A lattice is called orthocomplemented if it is equipped with an orthocomplementation. Such a lattice L is called orthomodular if moreover, for all x, y E L, x 5 y implies a: V (xL A y)

= y.

Since the orthocomplementation induces an isomorphism L + lent to x 5 y implies y A (yL V x) = x.

LO'

this is equiva-

Given two order-preserving maps f : PI+Pz and g: P2+PI in opposite directions, we say that f is a left adjoint of g, and g a right adjoint of f , written f -I g, if they satisfy one, and hence all, of the following equivalent conditions: 6

~ Birkhoff .

[I9671calls these 'atomic' or 'point lattices'.

Propositional Systems, Hilbert Lattices and Generalized Hilbert Spaces

i. ii. iii. iv.

for all x E PI and y E P2 we have f (x) 5 y a: 5 g(y), f(x) =min{y E P z I x = 7. This shows in particular that in a nonclassical orthomodular space H over K, if x is any nonzero vector, there is generally no vector u E K x such that < u, u >= lK, and that if y is a nonzero vector orthogonal to x there is generally no v E K y such that < u , u >=< x, x >.

(a),

---

Let us recall some basic facts of the theory of orthomodular lattices. For more , details the reader may consult [Kalmbach, 19831 or [ ~ t i k19901. The class of OMLs is the standard variety generalizing HLs (and GHLs). Many notions and properties of Hilbert lattices may be extended to OMLs. , (L, O , 1 , V, A) is a bounded An OML is'an algebra L = (L, O,1, V, A , ~ )where lattice, and is an antitone (i.e. such that a 5 b implies bL 5 a'-) and involutive unary operation so that a V aL = 1 (which implies a A aL = O), and the orthomodular law a V b = a V (aL A (a V b)) holds true. The class of OMLs is a variety containing the variety of Boolean algebras. Two elements a, b of an OML are said to be orthogonal (abb. a I b) if a 5 bL (or, equivalently, if b 5 aL). For any a, b in an OML, we will use the notation a + b for aL V (a A b), and cp,(b) for (b v aL) A a. Two elements a and b are said to commute each with other if the sub-OML

Equations and Hilbel-t Lattices

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generated by {a, b) is Boolean. This holds iff there exist a l , bl, c in L, pairwise orthogonal, such that a = a1 V c and b = bl V c. In particular if a and b are comparable, they commute. A triple (a, b, c) of elements of an OML such that a commutes with b and c is distributive, which means that, for any permutation (u, v, w) of (a, b, c), u A (v V w) = ( U A V V ) ( U A W )and u V ( v A w ) = (uVv)A (uVW). If D is a subset of an OML such that any two elements commute, the s u b OML generated by D is a Boolean algebra. A block of an OML is a maximal Boolean subalgebra. Any finite OML L may be represented by its Greechie diagram [Kalmbach, 19831, a hypergraph, whose vertices correspond to the atoms and whose edges represent the blocks of L. Here we will use such diagrams only in the simplest case of finite Greechie OMLs [~reechie,1971; Pthk, 1990], where any two distinct blocks of L have at most one common atom. Such Greechie diagrams are characterized by the fact that they admit no loop of order < 5. Two distinct blocks of a Greechie OML are called adjacent if they have a common atom. An OML whose all blocks have the same finite cardinal k is called k-homogeneous. Any finite 3-homogeneous OML is a Greechie OML. Let L be a finite Greechie OML, let D be its Greechie diagram and let B1,. . . ,Bp be some, but not all, of the blocks of L. By deleting in D the edges representing the blocks B1, . . . , Bp, and all the vertices representing the atoms not belonging to any block B # B1,.. . , Bp, we get the diagram of a new OML L'. We will say that L' is got from L by deleting (or removing) the blocks B1,. . . , Bp. The distance between two atoms a, b of a finite OML L is the least integer d such that there exists a finite sequence a0 = a, al, . . . , ad = b of atoms such that ai-1 Iai for i = 1,.. . d, if such a sequence exists, otherwise this distance is +m. Let us suppose that L is a finite Greechie OML, and that al,, . . ,a, are n atoms of L such that the distance between any two of them is at least 3. In this case (c.f. [Carrega, 1998]), a "prong of tzp p" pasted on a l , . . . ,a, is obtained by adding n 1 new atoms p, bl, . . . ,b, and n news blocks B1,.. . ,B, to the diagram of L, the atoms of Bi being ai, bi and p. When pasting a prong, we do not create any loop of order < 5, hence we get the diagram of a new OML. In the theory of OMLs, any inequality t 5 t' (where t, t' are two terms) is equivalent to the equality t = t A t'. Moreover, let X I , . . . ,x, be n variables, I a subset of {(i, j ) : 1 5 i < j 5 n), and E(xl,. .. ,x,) be an equation in the variables XI, . . . ,x, (which means that {xl, . . . ,x,} contains all the variables occurring in this equation). Then, in the theory of OMLs, the sentence ( V ( i ,j ) E I , x i I xj) + E(xl,. . .x,) is equivalent (c.f. [Mayet, 19861) to the equation E(t1,. . . t,), where tl = XI, and, for 1 < k 5 n, tk = xk A t i A . . . A tf with nk {il,. . . ,in,} = {i : 1 5 i < k and (i, k) E I). We will often write equations in this form. If E' is a set of equations in the theory of OMLs, we denote by Mod(&') the class of all OMLs in which hold all the equations of El. An equation E is called a consequence of &' if Mod(&') 2 Mod({E)), otherwise we say that E is independent of El. Let E and E' be two equations. An equation E is stronger than E' (or E'

+

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is weaker than E ) if Mod({E)) C Mod({E'}). These two equations are said to be equivalent if each of them is stronger than the other one. If E is stronger than E' and if they are not equivalent, we say that E is strictly stronger than E'. In the following, & will denote the set of all equations holding in all Hilbert lattices, but failing in some OML. A real-valued state on an OML C is a mapping s from C to the real closed interval [O,l], such that s(1) = 1, and a Ib + s(a V b) = s(a) s(b). An OML C is called unital if, for any a E C \ {0), there exists a real-valued state s on C such that s(a) = 1. This OML is called rich if, for any a, b E C, with a b, there is a state s such that s(a) = 1 and s(b) < 1. It is called brich if in the same conditions there exists a Zvalued state (i.e. a state with values in {O,l)) s such that s(a) = 1 and s(b) = 0. A mapping s from C to a Hilbert space H is called a Hilbert-valued state [Jajte and Paszkiewicz, 1978; Mayet, 1987; Mayet, 20061 if 11s(1)11 = 1, and for any a, b E C such that a Ib we have s(a) I s(b) and s(a V b) = s(a) s(b). For the t , we will only use here the states with values in a reasons given in [ ~ a ~ e20061, real Hilbert space, that will be called H-states. An OML C is called H-unital if there exists a real Hilbert space 7-t such that, for any a E C, a # 0, there is a H-valued state s on C such that IIs(a)ll = 1. The OML C is called H-rich if there exists H such that, for any a, b E C, with a $ b, there is a 'H-valued state s such that IIs(a)ll = 1 and IIs(b)ll # 1. If H is a real Hilbert space, then, for any unit vector u E 7.1, the mapping a I+ pr,(u) from C(H) to H is a H-state s, on C(H) such that for any b E C(H), s,(b) = u if€ u E b. Since for any a, b E C(7-t) such that a b there exists a unit vector u E a \ b it follows that C(H) is H-rich. If 7-1 is a nonreal Hilbert space, it t , that admits a natural structure of real Hilbert space, and it follows [ ~ a ~ e20061 C(7-t) is H-rich. If s is a H-state on an OML C, then, for any a E L, s(a) + s(aL) = s(1) and s(a) Is(aL), hence ll~(a)11~l l ~ ( a ~ ) 1=11, ~ which proves that IIs(a)ll 5 1 and that 11s(a)ll = 1 iff ~ ( a = ) s(1). This also shows that if s(a) I s(1) then s(a) = 0. Moreover, if a, b are any two elements of C such that a I b, then Ils(a V b)1I2 = ll~(a)11~11~(b)11~, which shows that the mapping a I+ ll~(a)11~ is a real-valued state on C. It also follows that if a 5 c then IIs(a)ll 5 IIs(c)II. In particular, if a 5 c and s(a) = s(l), then IIs(c)II = 1 hence s(c) = s(1). It also follows that if C is H-rich, then C is rich, and if C is H-unital, it is unital. Let us assume that sl is a 2-valued state on L. Then, if 71 is any nonzero real Hilbert space, and if el is a unit vector in H , by setting, for any a E C, s(a) = sl(a).el, we get a H-state. It follows that any 2-rich OML is H-rich.

+

+

+

+

3 ORTHOARGUESIAN EQUATIONS AND SOME OTHER ONES Let us remind the reader that & denotes the set of all equations, in the theory of OMLs, holding in every HL, but not in all OMLs.

Equations and Hilbert Lattices

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The first equation in E was found in 1975 by A. Day (unpublished, see [~odowski and Greechie, 1984; Greechie, 19811). This equation is called the orthoarguesian law, since it can be deduced from a particular case of the arguesian law. It will be denoted here by 0A2:

(a0 V bo) A (a1 V bl) A (a2 V b2)

I bo V (a0 A (a1 V [to,1A (t0,2 V t1,2)1))

(OA2)

where ti,j = (ai V aj) A (bi V bj). It was proved by R. Greechie [Greechie, 19811 and G. Kalmbach [Kalmbach, 19831 that there exists an OML in which it fails. Since that time, several equations in E have been got in a similar way. The first one [~odowskiand Greechie, 19841, denoted here by OA1 is strictly weaker [ ~ e g i land l Pavieit, 20001 than 0A2: (ai I bi, i = 0 , l ) =+ (a0 V bo) A (a1 V bl) L bo V (a0 A (a1 V to,i))

(0-41)

More recently, D.N. Megill and M. Pavieit [ ~ e g i land l PaviGt, 20001 have studied an infinite sequence (OAn)n21 of generalized orthoarguesian equations beginning with the above equations OAl and 0A2. If ti,j is defined as above, for any distinct integers i,j, then, for each integer n 1, the equation OA, may be written :

>

where the term s, is defined by induction: sl = toSland, for n 2 2, s, is obtained from 3,-1 by replacing in this term, for 0 i < j n - 1, all the occurrences of the subterm ti,j by ti,j A (ti,, V itj,,). By setting a, = 0 and b, = 1 in OA,, we get OAn-l, which proves that OA, is stronger than OA,-1. In [ ~ e g i land l Pavieit, 20001, it is proved that these equations hold in any HL, hence belong to E .




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Proof. (a) If X is any complex number of modulus 1such that An # 1, then both (L, Sx)and (L, SAn) are NOSOLs. Since ST,, = SAn,o,it follows that if we let S = SA,equation (i) is satisfied. If X # 1 is a pth complex root of unity, and S = Sx,then (ii) holds true. (b) In both cases, there is only one structure of NOSOL on L, which is its structure of OSOL (L,S). If n is an even, then St = le, hence, if a , x E C(7-l) do not commute each with other, we have x V pa(x) > x whereas x V (Sa)n(x) = x, which proves that (i) fails. If p is any odd number, then, since S,P= S,, equation (ii) fails. For instance, if p is an odd prime number, and if 7-l is a complex Hilbert space of dimension 2 3, there is, up to isomorphism, an unique structure of NOSOL (C(7-l),S), such that, for any a E L = C(X), (Sa)* = lL, the corresponding complex number X # 1being any pth complex root of unity. Let us observe that the different methods described in the other sections of this chapter do not allow to get any equation, in the language of OMLs, characterizing complex HLs among all HLs. The characterizations using isomorphisms of OMLs t , are clearly connected with the above ones. given in [ ~ a ~ e19981 A complex HL is equipped with several structures of NOSOL, with some interrelations between them, and this allows to define structures more complex than the previous ones in order to get better approximations of complex HLs by varieties [ ~ a ~and e t PulmannovL, 19941. 8 CONCLUDING REMARKS The feeling of the reader is probably that the search of equations holding in HLs is an endless problem. This is possible, since it is not excluded that the variety generated by HLs does not admit a recursive equational basis. On the other hand, this is far from being proved since, as far as we know, it is even not known whether or not this variety is finitely based. We have described in this chapter different methods for obtaining equations. But it is probably possible to find other ways, possibly more powerful. For obtaining equations, in some cases, we started from a finite OML which is not rich, or not H-rich. We observe that in each case, we have restricted ourselves to Greechie OMLs. Perhaps it is possible to get new and interesting equations by using more general finite OMLs and maybe by starting from other structures like those studied in section 7. The method using tensor products [ ~ a ~ e20061 t , may perhaps be improved and extended. Moreover, the above method for obtaining equations connected to real-valued states has been extended to equational implications [Mayet and PtLk, 1999; Mayet and Pt&, 20001, and probably such a generalization is also possible as concerns H-states.

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There are several motivations, besides the purely mathematical interest, for continuing this search. For example, it is not excluded that the knowledge of (some of) these equations could appear to be useful for the construction of algorithms in quantum computation. Furthermore, a basic problem in quantum logic, which has not been evoked in this chapter, is to find an algebraic structure more convenient than general OMLs for the study of quantum logic, on which could be defined a probability theory, in such a way that tensor products do exist. Although the ways explored until now use structures simpler than general OMLs, there is perhaps a chance to succeed in this search by studying the varieties (or quasivarieties) of OMLs containing Hilbert lattices, obtained by adding new axioms, and possibly new operations. For instance, as far as we know, nothing is known about the existence of tensor products of structures described in section 7, such as OSOLs or NOSOLs, to which could be possibly added some equations in E . It is perhaps possible t o find some equations in I failing in an OML C for which it has been proved that C €3C does not exist [Foulis and Randall, 19791, and then it would be appropriate to add such equations when searching a variety admitting tensor products. BIBLIOGRAPHY [Carrega, 19981 J. C. Carrega. Coverings of [Mon] in orthomodular lattices, Int. J. of Theoretical Physics, 37, 11-16,1998. [Carrega and Greechie, t o appear] J. C. Carrega and R. J. Greechie. Coverings of [MO,] and minimal orthonzodular lattices, Algebra Universalis, t o appear. [Carrega and Mayet, 20041 J. C. Carrega and R. Mayet. Orthomodular lattices from 3dimensional quadratic spaces, Algebra Universalis, 52,49-88,2004. [Chevalier, 19981 G. Chevalier. Orthosymmetries and Jordan triples, Int. J. of Theoretical Physics, 37,577-583,1998. [Foulis and Randall, 19791 D. J. Foulis and C. H. Randall. Tensor products of quantum logics do not &t, Notices of the A.M.S., 26, A-557,1979. [Godowski, 19811 R. Godowski. Varieties of orthonzodular lattices with a strongly full set of states, Demonstratio Math. X I V , 725-732,1981. [~odowski,19821 R. Godowski. States on orthonzodular lattices, Demonstratio Math. XV, 817822, 1982. [Godowski and Greechie, 19841 R. Godowski and R. J. Greechie. Some equations related to states on orthomodular lattices, Demonstratio Math. XVII, 241-250,1984. [Greechie, 19711 R. J. Greechie. Orthonzodular lattices admitting no states, Journal of Combinatorial Theory 10, 119-132.1971. [Greechie, 19811 R. J. Greechie. A non-standard Quantum Logic with a strong set of states, in Current Issues in Quantum logic, E. Beltrarnetti and B. C. van Raassen eds., Plenum Press, New York, 375-380,1981. [Gross and Kiinzi, 19851 H. Gross and U. M. Kiinzi. On a class of orthomodular quadratic spaces, L'enseignement m a t h h a t i q u e 31, 187-212,1985. [Hamhalter and Navara, 19911 J. Hamhalter and M. Navara. Orthosymmetry and modularity in ortholattices, Demonstratio Mathematica XXIV, 323-329,1991. [Holland, 19951 S. S. Holland, Jr. Orthonzodularity in infinite dinzension; a Theorem of M. Soldr, Bulletin of the AMS, New Serie 32,205-234,1995. [Jajte and Paszkiewicz, 19781 R. Jajte and A. Paszkiewicz. Vector measures on the closed subspaces of a Hilbert space, Studia Mathematica 63, 229-251,1978. [Kalmbach, 19831 G. Kalmbach. Orthomodular Lattices, Academic Press, New York, 1983. [Keller, 19801 H. A. Keller. Ein nichtklassischer Hilbertscher Raum, Mathematische Zeitschrift 172, 41-49,1980.

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[Keller, 19881 H. A. Keller. Measures o n orthomodular vector space lattices, Studia Mathematics 88, 183-195, 1988. [Mayet, 19851 R. Mayet. Varieties of orthomodular lattices related t o states, Algebra Universalis, 20, 368-396, 1985. [Mayet, 19861 R. Mayet. Equational bases for some varieties of orthomodular lattices related t o states, Algebra Universalis, 23, 167-195, 1986. [Mayet, 19871 R. Mayet. Classes Cquationnelles de treillis orthomodulaires et espaces de Hilbert, T h b e de doctorat d'Etat, Universitd Lyon 1, 1987. [Mayet, 20061 R. Mayet. Equations holding i n Hilbert lattices, Int. J . o f Theor. Phys., 45, 12161246, 2006. (N.B. B y mistake, this article has been published without correcting a list o f errors, and a corrected version is t o appear.) [ ~ a ~ e19921 t , R. Mayet. Orthosymmetric orfholattices, Proceedings o f t h e A.M.S., 14, N02, 295-306, 1992. [ ~ a ~ e19981 t , R. Mayet. Some characterizations of the underlying division ring of a Hilbert lattice by isomorphisms, Int. J . o f Theor. Phys., 37, 109-114, 1998. [Mayet et al., 20001 R. Mayet, M. Navara, and V . Rogalewicz. Orthomodular lattices with rich state spaces, Algebra Universalis, 43, 1-30, 2000. [Mayet and P t i k , 19991 R. Mayet and P. P t i k . Quasivarieties of orthomodular lattices determined by conditions on states, Algebra Universalis, 42, 155-164, 1999. [Mayet and P t i k , 20001 R. Mayet and P. Pt&. Orthomodular lattices with state-separated noncompatible pairs, Czech. Math. Journal, 50 (125), 359-366, 2000. [Mayet and Pulrnannovi, 19941 R. Mayet and S. Pulmannovzi. Nearly orthosymmetric orfholattices and Hilbert spaces, Foundations o f Physics, 24, NOIO, 1425-1437, 1994. [Megill and PaviBC, 2000] D. N. Megill and M. PavieiC. Equations and state and lattice properties that hold in infinite dimensional Hilbert space, Int. J . o f Theor. Phys., 39, 2337-2379, 2000. [Megill, 20031 D. N . Megill. Orthomodular lattices and beyond, Award-2003, Argonne workshop on automated reasoning and deduction, Chicago, July 10-12, 2003. iron, 19631 C. Piron. Foundations of quantum physics, Benjamin Inc., New Y o r k , 1963. [Pt&, 19901 Pt&, P. and Pulmannovi, S., Orthomodular Structures as Quantum Logics, Kluwer Academy Pub., Dordrecht, 1990. [ ~ u l m a n n o v i19941 , Pulmannovii, S., Quantum logics and Hilbert spaces, Found. o f Physics 21, 1403-1414, 1994. [Pulrnannavi, 19991 S. PulmannovB. Aziomatization of quantum logics, Int. J . o f Theor. Phys., 35, 2309-2319, 1999. [Solbr, 19951 M. P. Sol&. Characterization of Hilbert spaces by orthomodular spaces, Comrnunications i n Algebra 23, 219-243, 1995. [Varadarajan, 19841 V . S. Varadarajan. Geometry of Quantum Theory, Springer-Verlag, NewYork, 1984.

HANDBOOK OF QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM STRUCTURES Edited by K. Engesser, D. M. Gabbay and D. Lehmann O 2007 Elsevier B.V. All rights reserved

555

THE SOURCE OF THE ORTHOMODULAR LAW John Harding 1 INTRODUCTION

Beginning with Birkhoff and von Neumann [1936], a central theme in quantum logic is to consider generalizations of the lattice C(7-l) of closed subspaces of a Hilbert space 7-l as models for the propositions of a quantum mechanical system. Husimi [Husimi, 19371 was the first to note that the ortholattices C(7-l) satisfied the following identity known as the orthomodular law:

This fact was rediscovered several times in the 1950's and 1960's [Kaplansky, 1970; Loomis, 1955; Mackey, 1963; Maeda, 1955; Piron, 19761 and lead to the role of orthomodular lattices (abbreviated: OMLS)and orthomodular posets (abbreviated: OMPS)as abstract models for the propositions of a quantum mechanical system. It is instructive to see how the validity of the orthomodular law in C(7-l)follows in a transparent way from basic properties of Hilbert spaces. ,For the non-trivial containment in (1) note that if b E B, then b = a1 +a2 for some unique a1 E A and a2 E AL. Then if A G B we have a1 E B, hence b - a1 = a2 belongs to AL n B, and therefore b = a1 a2 belongs to A V (AL A B). Thus, the validity of the orthomodular law in C(7-l) follows as each vector in 7-l can be uniquely expressed as a sum of vectors from A and A ~ This . shall be of fundamental importance to

+

us. The orthomodular law has several equivalent formulations that highlight different aspects of its nature. We mention one of these that provides insight that will be helpful here. Note first that every ortholattice L is equal to the set-theoretic union of its Boolean subalgebras as each a E L lies in the Boolean subalgebra (0, a, a', 1). Orthomodular lattices are exactly those ortholattices L where the partial ordering of L is determined by the partial orderings of its Boolean subalgebras. In this sense, OMLS are exactly the locally Boolean ortholattices. While the above comments make a case that the orthomodular law is worthy of study, the over fifty year longevity of the orthomodular law must be attributed to its role in developing a large body of deep and beautiful mathematics and to its role in the theoretical foundations of quantum mechanics. We cannot describe these results in any depth, but point to the work on the dimension theory of projection

556

John Harding

lattices [ ~ u r r and a ~ von Neumann, 19361 and the subsequent development of a dimension theory for certain OMLS [~oomis,1955; Maeda, 19551; the related development of continuous geometry [von Neumann, 19601 and its deep ties to modular ortholattices [Kaplansky, 19551; the work on Baer*-semigroups [Foulis, 19601; the beautiful algebraic theory of OMLS outlined in [Bruns and Harding, 2000; Kalmbach, 19831; generalized Hilbert spaces o iron, 1976; Keller, 19801 and Solcr's theorem [Solkr, 19951; generalized orthomodular structures [~ouliset al., 1992; Greechie and Foulis, 1985; KGpka, 19921 and their ties to partially ordered abelian groups [Foulis and Bennett, 19941; and of the multitude of work on applications of orthomodularity to the foundations of quantum mechanics of which [Beltrametti and Casinelli, 1981; Gudder, 1988; Mackey, 1963; Piron, 19761 is a sample. As a meta-question, one might well ask why such an innocent looking identity as the orthomodular law should lie at the heart of so much interesting mathematics. While the orthomodular law has its origins and many of its application centered on Hilbert spaces, we present here a very different view of orthomodularity, one that eliminates the reliance on Hilbert spaces and focuses on a more elementary mathematical property instead. We present our thesis in an informal manner below, and provide a more detailed treatment in the body of this chapter. The crucial item is the following.

Slogan I: The direct product decompositions of many familiar mathematical structures, including sets, groups, modules, and topological spaces, naturally form an orthomodular poset. Thus, orthomodularity at its root arises from considering direct product decompositions. Of course, we will make this precise in the sequel, in particular in Theorem 13. The point of this statement is to identify decompositions as the basic mathematical process that leads to orthomodularity. What then about the about the intimate link between orthomodularity and Hilbert spaces?

Slogan 11: Orthomodularity has nothing to do with Hilbert space, it is a consequence of considering direct product decompositions of a Hilbert space. Here we mean orthomodularity is not a property of a Hilbert space 7.1, but arises only when we consider the lattice C(H) of closed subspaces of H. As we noted above, the key property of Hilbert spaces is that for a closed subspace A, each vector u in H can be uniquely represented as the sum of a vector U A E A and a vector U A L E A'-. Viewed another way, this means that each closed subspace A induces a direct product decomposition H A x AL of H and all direct product decompositions of H arise in this manner. Thus the process of considering the closed subspaces of H is really a disguise for considering the direct product decompositions of 8.

--

Slogan 111: The role of orthomodularity in the foundations of quantum mechanics is due, at least in part, to an underlying role of direct product decompositions in the foundations of quantum mechanics.

The Source of the Orthomodular Law

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In support of this statement, we will give an axiomatic development of what we term an experimental system based essentially on the notion of direct product decompositions. This will include the development of a system of yes-no experiments, a type of logic for such experiments, a probabilistic treatment of the results of such experiments, and an approach to observables and their calculus. The standard Hiibert space approach to quantum mechanics will be shown to fit exactly into this framework.

Slogan IV: Independent of any connection to quantum mechanics, orthomodularity is worthy of study due to its role in the theory of direct product decompositions. This chapter is organized in the following manner. In the second section we review well-known facts about the treatment of surjective images of structures such as sets and groups. This is a familiar topic that is presented from a perhaps unfamiliar viewpoint. The treatment we give provides a sort of toy model for our treatment of direct product decompositions. In Section 3 we begin our study of direct product decompositions of sets and prove our Main Theorem - that the binary direct product decompositions of a set, with operations induced in a natural way, forms an OMP. In the fourth section we illustrate this result with several concrete examples, and in the fifth section we extend our results on decompositions of sets to decompositions of other types of mathematical structures, such as groups, modules, topological spaces, Hilbert spaces, and so forth. A number of well-known methods to construct OMPS arise as instances of this construction. Section 6 contains a finer study of the structure of the OMPS BDec X of decompositions of a set X. In particular, we characterize the Boolean subalgebras of such OMPS. The finite Boolean subalgebras with n atoms are shown to correspond to direct product decompositions of X with n factors, while the infinite Boolean subalgebras are shown to correspond to a certain type of continuously varying direct product decomposition known as a Boolean sheaf (or Boolean product). This characterization of the Boolean subalgebras of BDec X leads to a complete description of compatibility in such OMPS, and the result that all such OMPS are regular. These results are exploited in numerous ways in the subsequent section, and in particular lead to a type of logic based on decompositions. In section 7 we give an axiomatic presentation of what we term an experimental system focused very tightly on the notion of direct product decompositions. Roughly, the key ingredient is the requirement that each n-ary experiment on a system induces an n-ary direct product decomposition on the state space of the system. From this we obtain that the binary experiments (the secalled questions of the system) form an OMP,and that this OMP of questions has a sane interpretation of its logic. With a few more axioms we have the notion of an experimental system with probabilities. From this we develop an interpretation of the probabiiity of an experiment yielding a certain result when the system is in a given state, and then a theory of observables, their expected values, and their calculus. The

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standard Hilbert space model for quantum mechanics is seen as a specific instance of such an experimental system. The material in these first seven sections provides enough of the core work on decompositions for the reader to gain a good feel for the subject. A number of additional results not necessary for a first view are given briefly in Section 8 along with a number of open problems for the reader who wishes to pursue the subject. A brief conclusion is then presented as Section 9. This chapter is presented as an invitation to those who may wish to further explore the subject. It is written in a somewhat informal way, and includes only a very few proofs whose purpose is to better illustrate the nature of the mathematics involved in the underlying theory. This subject has unfortunately almost entirely been my own project, and the results here are (nearly) all contained in the papers [Harding, 1996; Hardiig, 1998; Harding, 1999; Harding, 2001; Harding, to appear] where complete proofs can be found. 2 SURJECTIVE MAPS AND QUOTIENTS

In this section we review some basics about surjective (onto) mappings. While this material is in some way familiar to us all, it may familiar be at a more subconscious level, and a more organized treatment of these ideas may not be immediately at hand. As our treatment of decompositions will in many ways mirror our treatment of surjective mappings, this section serves as both a review and a preview for later developments. DEFINITION 1. Given a set A, a surjection with domain A consists of a set B and a surjective (onto) mapping f : A -+ B. When considering surjections with domain A, or any type of mapping for that matter, one is often not interested in the particular nature of the elements involved, but only in the way elements are transformed into others. We make this precise in the following. DEFINITION 2. Define an equivalence relation = on the collection of all surjections with domain A by setting f : A + B to be =related to g : A -+ C if there is an isomorphism (bijection) i : B -+ C with i o f = g.

We use [f : A + B] for the =-equivalence class of the surjection f : A + B, and we let Surj A be the set of all equivalence classes of surjections with domain A.

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Putting structure on the set Surj A is key. DEFINITION 3. Define 5 on Surj A by setting [f : A + B] 5 [g : A t h e r e i s a m a p h : B - t C w i t h h o f =g.

4

C] if

Of course, one must verify that the above definition of 5 is independent of the representatives of the equivalence classes involved, but this is routine. Our aim is to prove the following. THEOREM 4. ( S u r j A, 5 ) is a complete lattice. Here we could proceed directly, using obvious arguments to show 5 is reflexive and transitive, and the fact that surjections are epic for anti-symmetry, to obtain that 5 is a partial ordering. The existence of arbitrary joins can be obtained from properties of products, and then the existence of meets follows from general principals. However, it is common to treat surjections by showing ( S u r j A, 5 ) is dually isomorphic to the lattice of equivalence relations on A. In any event, one must establish this dual isomorphism as it is necessary to make computations tractable. DEFINITION 5. For f : A -t B define kerf = {(x, y)I f (x) = f ( y ) ) . THEOREM 6. The structure ( S u r j A, 5 ) is dually isomorphic to (Eq A, C), the set of equivalence relations on A partially ordered by set inclusion, via the map that takes [f : A + B] to kerf. One uses this dual isomorphism and the following description of joins and meets in the lattice of equivalence relations to effect computations with surjections. PROPOSITION 7. For A a set, (Eq A, C) is a complete lattice where

1. Meets are given by intersections.

2. Joins are given by the transitive closure of the union. One additional detail regarding computations in Eq A will be important. We recall that for relations 8,q5 on A that the relational product 8 o q5 is the relation {(x, z ) lx8y and y4.z for some y). The following is well known [Burris and Sankap panavar, 19811. PROPOSITION 8. For 8,q5 equivalence relations on A these are equivalent. 1. 8 o 4 = q5 o 8, i.e. 8 and q5 permute.

To summarize, we believe that the lattice ( S u r j A, 5 ) is the primitive notion. The more familiar (Eq A, c) arises as a means to effectively work with this primitive notion. Essentially, from each equivalence class of surjections [f : A + B], is the we choose a canonical representative Ke : A + A/8 where 8 = kerf and natural quotient map. Then rather than study the surjections K e : A + A/8, we simply study the equivalence relations 8 that determine them.

John Harding

3 DECOMPOSITIONS This section contains the core material around which the chapter is built - the notion of a direct product decompositions of a set. Our treatment will mirror the treatment of surjections given in the previous section. DEFINITION 9. A direct product decomposition, or simply a decomposition, of a set A consists of a finite sequence of sets Al, . . .,A, and an isomorphism f :A+A1x-..xAn. When the map f is clear from the context, and cumbersome to write, we use A cx A1 x x A,. We need some notation for working with maps and products. DEFINITION 10. Suppose A, A1, . . . ,A, are sets, f : A 4 Al x . . . x A, and that gi : A 4 Ai for eacxh i = 1,.. . , n.

1. Define

: A1 x

. . -x A,

+ Ai

to be the ith projection map.

2. Define fi : A 4 Ai to be the composite .rri o f .

. x A, we have f = fl x . .. x f,. Just as with surjections, when considering a decomposition f : A 4 Al x . - .x A, one is often not interested in the particular elements of the sets Al,. . . ,A,, but only in how the bijection f maps elements of A into the elements of the product (for instance, which elements of A are mapped to ones with identical first components). This is made precise by the following. Thus for f : A + A1 x

DEFINITION 11. Define an equivalence relation w on the decompositions of A by setting f : A 4 A1 x - .- x A, to be w-related to g : A 4 B1 x - - .x B, if m = n and there are there isomorphisms il, . . . ,in with ik : Ak -+ Bk and (il x . . - x in) o f = g.

Then [f : A Al x ... xA,.

4

A1 x . . - x A,] denotes the =-equivalence class of f : A

As with Surj A, it is key to place structure on the equivalence classes of decompositions. We begin by putting structure on the equivalence classes BDec A of binary decompositions of a set A, though later we will also place structure on the collection of all equivalence classes of decompositions. At heart, we use the

The Source of the Orthomodular Law

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obvious connection between A1 x A2 and A2 x Al to define a unary operation on BDec A, and we use the relationship between A1 x (A2 x A3) and (Al x A2) x A3 to define a relation I on BDec A. Before giving the precise definitions, we review a few facts about decompositions and the corresponding notation. The decomposition f : A + A1 x A2 is literally equal to fl x f2 : A -+ A1 x A2 (as f = fl x f2 by Definition 10). One sees that f 2 x fl : A -+ Az x A1 is also a decomposition. However, this decomposition is not even =-related to the original! This fact seems unusual as there is an obvious isomorphism i : Al x A2 + A2 x Al. But to have these decompositions %-related, Definition 11requires a pair of isomorphisms ill i 2 making the following diagram commute, and clearly this is not the case.

Note aslo that from a ternary decomposition f : A -+ Al x A2 x A3 we can build several binary decompositions such as fl x (f2 x f3) : A + A1 x (A2 x A3) and the quite different (fl x f2) x f3 : A + (A1 x A2) x A3. Again, there is an isomorphism between A1 x (Az x A3) and (A1 x A2) x A3, but this is certainly not sufficient to make these decompositions %-related. DEFINITION 12. For a set A, let BDec A be the collection of all equivalence classes of binary decompositions of A. Define a unary operation * on BDec A by setting

[f : A + A1 x A ~ ] = * [f2 x And define

5 on BDec A by setting [f : A

fl

-+

: A + A2 x All.

A1 x A2] 5 [g : A -+ B1 x Bz] if

for some ternary decomposition h : A -+ Cl x C2 x C3. The crucial definition of the relation 5 can be expressed in a somewhat different way. Whenever A is isomorphic to a ternary direct product C1 x C2 x C3, then the equivalence class of the binary decomposition A 11 C1 x (C2 x C3) is 5 that of the decomposition A 11 (Cl x C2) x C3. In a sense, it is this careful treatment of commutativity and associativity, or rather the lack of these properties, that gives our structure. We now present our primary result. THEOREM 13. For A a set, (BDec A, 5,*) is an

OMP.

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To prove S u r j A is a lattice, and to find tractable methods to compute in this lattice, we showed each equivalence class of surjections has a canonical representative A t AIO uniquely determined by an equivalence relation on A. Computations in S u r j A are then reduced to computations with equivalence relations. We now follow a similar path to show B D e c A is an OMP, and to give tractable methods to compute in this OMP. DEFINITION 14. Two equivalence relations 0, q5 on a set A permute if Oorj = 400. A set of equivalence relations on A is pairwise permuting if any two members of the set permute. A set of equivalence relations on A is called a Boolean subsystem of Eq A if it is pairwise permuting and forms a Boolean sublattice of the lattice Eq A. As we will see, Boolean subsystems are closely linked to direct product decompositions. We require one further definition. DEFINITION 15. A factor n-tuple of a set A is a finite sequence (81,. . . ,On) of equivalence relations on A whose members that differ from the universal (largest) relation V on A are distinct and comprise exactly the coatoms of a Boolean subsystem of Eq A. So a factor pair is an ordered pair (01, 82) of permuting equivalence relations that are complements in the lattice Eq A. A factor triple (81, 02,03) is formed either by inserting V into a factor pair, or by taking the coatoms of an eight-element Boolean subsystem as shown below.

The following result is the key link between factor tuples and decompositions. s In the binary case it is well known and easily found in the literature [ ~ u r r i and Sankappanavar, 19811. The general case follows easily, and is found in [~cKenzie et al., 1987, pg. 1611. PROPOSITION 16. Let A be a set.

I . I f f : A 4 Al x . . . x An i s an n-ary decomposition, then for Oi = ker fi we have that (81,. . . ,On) i s a factor n-tuple of A.

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2. If (01, . .. ,en) is a factor n-tuple, then the natural map KO, x . . an n-ary decomposition A A/& x . . x Ale,.

-

--

XKe,

provides

This result implies that each equivalence class [f : A -+ A1 x - .. x A,] has A/& x - .. x A/& and this leads to a bijective a canonical representative A correspondence between equivalence classes of n-ary decompositions and factor ntuples where [f : A -,A1 x . .x A,] corresponds to (ker fl, . . . ,kerf,). So instead of working with the set BDec A of equivalence classes of binary decompositions of A, we can work instead with the set of all factor pairs (81, 02) of A. For this to be advantageous, the structure on BDec A, particularly the relation 5, must have a tractable description in terms of factor pairs. Fortunately, this is the case. PROPOSITION 17. Let (01, 02) and (41, $2) be factor pairs of A. equivalent.

I . In BDec A we have [A 1: A/O1 x Ale2] 5 [A

--- A/&

x A/42].

2. 01, 02, 41,42 belong to a Boolean subsystem of Eq A and 3.

41 C 81, 02 C 4 2

These are

C 81.

and 41 o02 = e2 041.

A proof of this result is found in [Harding, 1996, Lemma 3.31. It provides a very simple method to work with the relation 5,something that seems quite intractable when dealing directly with decompositions. DEFINITION 18. For a set A, let Fact A be the set of all factor pairs (81, 82) of A. Define a unary operation * on Fact A by setting

and define a relation 5 on F a d A by setting (01,02) 5 (41,452) if

41 c el, e2 c 41 and 41 0 e2 = e2 0 41.

When convenient, we refer to the structure (Fact A, 5 ,*) simply as Fact A. The manner in which the above structure is defined on Fact A, in conjunction with Proposition 17, then immediately provides the following result, which is of course, the reason we are working with the structure Fact A in the first place. PROPOSITION 19. For A a set, (BDec A, 5 , *) is isomorphic to (Fact A, 5,*). Due to this isomorphism, the following result obviously provides our primary objective of showing that (BDec A, 5,*) in an OMP. THEOREM 20. For a set A, the structure (Fact A, 5,*) is an

OMP.

Proof. We refer t o [Harding, 1996, Theorem 3.51 for a complete proof in a more general setting, but we will show that 5 is a partial ordering as this illustrates the nature of the arguments involved and may encourage the reader to fill the missing details themselves.

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Note that the definition of a factor pair shows 5 is reflexive, and anti-symmetry follows trivially from the definition of 5. For transitivity, suppose (el, 82) 5 (41,42) and (41,42) 5 ($1, $2). Then $1 C 41 G 81, 62 C 4 2 C $2, and both +2) we must show G el and 41~62and $1,452 permute. To show (81, 02) 5 82 C $2, which are obvious, and that $1, 82 permute. Suppose (a, b) E $1 082. As $1 G 41 we have (a, b) E 082, and as 82 C #JZ we have (a, b) E $1 0 4 2 . As 41,& and $1, 4 2 both permute, we have (a, b) E 82 o 41 and (a, b) E 4 2 0 $1. So there are c, d with ae2cq!qb and a+2d$lb. Then as 82 C 42 we have c42a42d so c42d, and as $1 C q51 we have cq51bq51d, so c&d. Then (c, d) belongs to 41 n 42, which is the diagonal relation A, so c = d. Thus aO2c = d$lb, showing (a, b) E 82 o $1. So $1 o 82 C O2 o $1 and a similar argument shows the other containment. H Above we have seen how to tractably work with * and 5 in Fact A. We next discuss computation of orthogonal joins @ in Fact A and BDec A. Later, in Section 6, we discuss compatibility in these structures. PROPOSITION 21. For a set A, two elements (el, 82) and (41, A) of Fact A are C el, 02 C $1 and 02, 42 permute. In this case, orthogonal if, and only if,

Two elements of BDec A are orthogonal if, and only if, they are of the f o m [A 21 Al x (A2 x A3)]

and

[A = A3 x (A1 x A211

for some ternary decomposition A E Al x A2 x A3. In this case, their orthogonal join is given by [A (A1 x A3) x AZ]. In sum, we have treated decompositions much the way we treated surjections. We gave a definition of decompositions, and defined an equivalence relation on the collection of all decompositions. We then put structure on the collection BDec A of equivalence classes of binary decompositions of A. To prove this structure gave an OMP and to provide tractable methods to work with this OMP we passed to an auxiliary set F a d A built from equivalence relations on A. The key idea is that each equivalence class of decompositions has a canonical representative that can be described in terms of equivalence relations. 4

SURJECTIONS AND DECOMPOSITIONS FOR FINITE SETS

Here we provide some concrete examples decompositions to give the reader a more definite feel for the subject. We begin with one small example of S u r j A, primarily to emphasize a certain point that sometimes causes difficulty. EXAMPLE 22. Consider the 3-element set A = {a, b, c). There are 5 equivalence relations on A; the diagonal relation A, the universal relation V, and the relations 62 = A U {(ale), ( ~ , a ) ) ,and 83 = A u {(b,~),(c,a)). So 61 = A u {(a, b), (b, the lattice Eq A, which is dually isomorphic to S u r j A, is as depicted below.

The Source of the Orthomodular Law

A point to emphasize i s that a n equivalence class of surjections [ f : A + B] is not simply determined by the cardinality of the image set B, but depends also o n how elements of A are collapsed together into elements of B. I n this case there are 3 distinct equivalence classes [ f : A -+ B] where B i s a 2-element set as there are 3 ways to choose a pair of elements of A to collapse together. Before giving examples of Fact A, we note that if A is a finite set and 0 belongs to a factor pair of A, then 8 is a regular equivalence relation, which means that each equivalence class (also called block) of 8 has the same number of elements. Thus if (61, 62) is a factor pair of A, then A is isomorphic to A/O1 x Ale2, and the number of blocks of O1 is the cardinality of Ale1, while the number of elements in each block of O1 is the cardinality of A/& as both are given by the cardinality of A divided by the cardinality of Ale1.

EXAMPLE 23. Suppose that A i s the 4-element set { a , b, c, d ) . Then there are 5 regular equivalence relations o n this set; A, V , the equivalence relation O1 whose blocks are { a ,b ) , { c , d}, the equivalence relation O2 whose blocks are { a , c ) , {b, d ) , and the equivalence relation O3 whose blocks are { a , d ) , {b, c). Each of the 6 pairs ( B i , O j ) where i # j yields a factor pair in this case, and it follows that Fact A i s the height two O M P M 0 6 shown below.

To reiterate, there are 8 equivalence classes of decompositions of a 4-element set. There i s one as the product of a 4-element set and a 1-element set, one as the product of a 1-element set and a 4-element set, and 6 digerent decompositions as the a product of two 2-element sets.

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EXAMPLE 24. Suppose A i s the 6-element set { a , b, c, d, e, f } . Aside from A, V , each regular equivalence relation 8 will have either two or three blocks, depending on whether A18 i s a two or three-element set. Factor pairs (81,82) not involving A, V will consist of one regular equivalence relation with two blocks and one with three. But not all such pairs of regular equivalence relations will provide a factor pair. For the relation 81 with blocks { a , b, c ) , { d , e, f } , 8~ with blocks { a , b), {c, d ) , { e , f), and 83 with blocks { a , d ) , {b, e ) , {c, f ) , we can see that (el,82) i s not a factor pair as 81 n 82 # A, while (81, 83) i s a factor pair. Again, the OMP F a d A will be a n MO, for some finite n (the combinatorial exercise of finding n i s left to the reader). At this point it may seem that the OMPS Fact A are of limited interest as each of in the previous two examples is of height two, consisting of blocks (maximal Boolean subalgebras) having exactly two atoms each. This however, is because each of the primary decompositions 4 = 2 x 2 and 6 = 2 x 3 involves exactly two prime factors. The following description of Fact A for a finite set A follows from results in Section 6. OMPS

PROPOSITION 25. Suppose A is an n-element set where n = p l . p z . . .pk i s the primary decomposition of n. Then the OMP Fact A i s of height k (the number of factors in the primary decomposition of n ) and consists of blocks all hawing k atoms each. It is useful to consider things from the perspective of B D e c A rather than Fact A. EXAMPLE 26. Let A = ( 0 , l I 3 and let f , g : A -t ( 0 , 1 ) x ( 0 , 1 ) x {O,1) be defined by f ( 2 ,y, z ) = ( x ,y, z ) and g ( x , y, z ) = ( x y, y, z ) , where addition i s modulo 2. These two ternary decompositions give rise to the following binary decompositions

+

B y definition, D 5 F and E 5 G . Surely D # E as there can be n o isomorphism il : { 0 , 1 ) -+ { 0 , 1 ) with il o fl = gl as this would require i l ( x ) = x y for all x, y, z E ( 0 , 1 ) . But F = G as the isomorphism jl : { 0 , 1 } x {O,1) -+ {O,1) defined by j l ( x , y ) = ( x + y, y ) satisfies jl o ( f l x f 2 ) = 92 xg2, and the isomorphism j2 : { 0 , 1 } -t {O,1) defined by j2 = i d satisfies j2 o f3 = g3. One then obtains that D , F generate one of the 8-element blocks of B D e c A, that E, G generate another distinct block, and that these two blocks have in common the coatom F = G and the atom F* = G*. I n particular, for this 8-element set A, the OMP B D e c A i s of height three and consists of 8-element Boolean algebras linked in a rather intricate way. The OMPS F a d A for A a finite set are combinatorially very interesting structures with many interesting symmetries. Of course, the structure F a d A for A

+

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an infinite set is surely of primary interest, as are sub-OMPSof Fact A induced by placing various types of structure on A and restricting to decompositions that respect this structure. This shall be the focus of the following section.

5 DECOMPOSITIONS OF SETS WITH STRUCTURE The term "set with structure" is a broad one, meant to include such familiar objects as groups, rings, vector spaces, and other algebras with finitary or infinatary operations; relational structures such as posets and graphs; topological structures and uniform spaces; and so forth. Our aim is to consider structure preserving decompositions of such objects. As before, it is instructive to begin with surjections. DEFINITION 27. For a group 8 , a surjection of B consists of a group li and an onto homomorphism f : G + li. An equivalence relation x is defined on the surjections of G as in Definition 2 and a relation 5 and is defined on the equivalence classes of surjections of G as in Definition 3. Of course, a similar definition would apply for any other type of algebra A, such as a ring, vector space, monoid, and so forth. Key to studying ( S u r j A, 5) for an algebra A is the correspondence between surjections of A and congruences of A (certain equivalence relations on the underlying set of the algebra that are compatible with the basic operations of the algebra [ ~ u r r i and s Sankappanavar, 19811). PROPOSITION 28. Given an algebra A with underlying set A, the structure ( S u r j A, 5 ) i s dually isomorphic to the sublattice ( C o n A, 5 ) of ( E q A, 5 ) consisting of all congruence relations on A. In specific settings, there are alternate, but equivalent, app'roaches to working with the lattice of surjections. For example, for groups one often works with the lattice of normal subgroups, for rings the lattice of ideals, and for vector spaces the lattice of subspaces. We turn now to decompositions. DEFINITION 29. A decomposition of a group consists of a finite sequence 81,.. .,Gn of groups and a group isomorphism f : 6 + x ... x Gn. We define an equivalence relation = on the decompositions of as in Definition 11 and we define a unary operation * and binary relation 5 on the collection BDec of all equivalence classes of binary decompositions of G as in Definition 12. Of course, similar definitions apply for rings, vector spaces, or indeed any type of algebra. A proof of the following is found in [Harding, 1996, Remark 4.21. PROPOSITION 30. For an algebra A with A as its underlying set, ( B D e c A, 5,*) i s isomorphic to the sub-OMP (Fact A, 5,*) of (Fact A, 5,*) consisting of all factor pairs (01, 02) where both 81,82 are congruences. For relational structures and topological spaces, the situation is similar (see [Harding, 1996, Theorem 4.4 and 4.61). A decomposition consists of a finite sequence of relational structures or topological spaces, and an isomorphism or home-

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omorphism, as the case may be, betweem the original structure and the product of this family of structures. Exactly as above, one obtains a structure ( B D e c A, (Zl), A2 : Mlf ( 0 ) t M: (E2) admit a joint observable. Indeed, if X1 and X2 are measurable subsets of El and Ez respectively, consider the real-valued function on the Dirac measures on R defined by the product A16,(X1). A26,(X2), and extend it over M ~ ( Rby ) affinity. In this way we get a function from M$(R) into the set of all rectangles X1 x X2, which extends to a measure on the Cartesian product El x Z2 by standard procedures (see, [ ~ i l l i n ~ s l1979, e ~ , Theorem 11.31). The map from M:(R) into M T ( 5 x22) SO obtained, to be denoted A1 xAz, is afFine and regular, thus being an observable on M? (0). Moreover, it is easy to

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check that

A1 and A2 are marginal projections of A1x A2,namely

This qualies A1 x A2 as a joint observable of A1 and A2 (for more details see [~eltramettiand Bugajski, 1995 b]). With some abuse of notation we are using the symbol x both for the Cartesian product (when applied to sets) and for the product observable. Let us remark however that, while in the standard classical case the two obA2 admit A1x A2 as the only possible joint observable, in the more servable~A1, general operational framework, where unsharp observables are allowed, A1 x A2 need not be the only possible joint observable of A1 and A2,and one may be faced with a plurality of joint observables, as simple examples can show. 6 THE BELL EFFECT

6.1 Consistent and complete collections of probability measures Consider a finite collection of disjoint measurable spaces El, ....,En:we shall later think of them as the outcome spaces of corresponding observables of some physical system. Consider also the hierarchy obtained by Cartesian products of them: from a first level containing the products Eix Ej,i # j= 1,...,n,to a second level containing the products of three factors, up to the last level where we have the ....,Znwill be called the base product Elx ..... x En.The measurable spaces El, elements of the hierarchy. As an example, let us visualize in Fig.1the n = 4 case:

-=I X - X - X =2

Z 1x

C3

C4

Z2x E3 Elx E2x E4 Elx E3x E4

Elx z2

E2x E3 x E4

- x E3 - - - - - - - - - - X =4

=1

1

=2

C 2 X =3 C3

=2 X e4

Z 3 X =4

&4

Figure 1. The n = 4 hierarchy Suppose now that on each measurable space Zi, i = 1,....,n,and on some of the product spaces of the hierarchy said above (not necessarily on each element of the hierarchy) a probability measure is defined: we write vi for the probability vi,j for the probability measure on Eix Ej (if it is defined), and measure on Zi, so on. Let us write M , for this collection of probability measures. We shall further suppose that Mnis consistent, in the sense that if it contains vil,...,ir on =dl x...xZirand vjl,...,json Ejlx...xEjswith {il, ..., iT} {jl..., , js} (1,...,n} then vil,...,ip is a marginal projection of vj,,..., j, in the usual sense of

-

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measure theory. As an explicit example, if the collection of probability measures contains vi,j then vi(Xi) = vi,j(Xi x Ej) for every Xi E B(Ei), and vj(Xj) = x Xj) for every X j E B(Ej). vi When the collection M, contains a measure vl,,.,,, on El x .... x En then this measure generates all the members of the collection by the mechanism of marginal projections, and Mn is said to be complete. If the consistent family M, is not complete, namely if it does not contain a measure on =1 x .... x En, then the question arises whether it can be made complete by the addition of a suitable measure on Z1 x ..... x Z,, namely whether there exists a probability measure vl,...,, such that Mn U {Y,...,,} is a complete exists , then it generates all the consistent family of measures. If such a y,..., members of M, by the mechanism of marginal projections, and we say that M , admits a generating measure. It is a relevant fact that there are consistent collections of probability measures which do not admit a generating measure: in other words they cannot be thought of as sub-collections of complete collections. When referred to the basic elements vl, ..., v, of M, (or to some of them) the definition of generating measure reduces to the familiar definition of joint measure. Notice that the set of basic measures y,...,v, always admits at least one joint measure. However, a joint measure need not be a generating measure of the consistent family Mn when the latter contains other measures beyond the basic elements. Consider for instance the joint measure of y,..., vn having the product form, namely the measure on Z1 x ..... x En which at the rectangle XI x ..... x X, (where Xi E B(Ei), i = 1,....,n) takes the value vl(X1) - .... . vn(Xn): such a joint measure is a generating measure of Mn only if all the non-basic elements of M, have the form of products of basic elements. Indeed, a marginal projection of a product measure is still a product measure. It should also be stressed that a generating measure of a consistent family of measures M, need not be unique (and when there are more than one then there are infinitely many because any convex combination of two generating measures is still a generating measure). However, we have the following

-

THEOREM 1. If for any {il,. . . ,i,} C (1,. . . ,n} the set {vi,, . . . , vip) of basic measures possesses only one joint measure then Mn admits a unique generating measure. This follows from the fact that a generating measure is also a joint measure of the basic elements, so that the uniqueness of the latter entails the uniqueness of the former. For a more complete account of this matter we refer to [Beltrametti and Bugajski, 19961.

6.2 Events and classical representability of probabilities The existence of a generating measure may imply certain numerical constraints on suitable linear functions of the values assumed by the members of M,, so that the

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violation of such constraints would become a criterion to exclude the existence of a generating measure. In order to make this fact more explicit, consider again the n=4 hierarchy of Fig.1 and take a collection of probability measures of the form

Further suppose that each of the measurable spaces Ei, i = 1,2,3,4, is a two-point set, say {ti,Sf}. Then the existence of a generating measure ~ 1 , 2 , 3 , 4would, for instance, imply

stand for the more correct but pedantic (the notations vi(ti) and vi ({ti}) and vi,j({C,G } ) ). Indeed, considering the 16-point space El x z2x E3 x and looking at the subspace consisting of the 8 points (t:,t;,t;, t i ) , (ti, t;, t;, ti'), (ti, t;, ti,t3, (ti, t;, ti', ti'), (