Unbounded Non-Commutative Integration
Non-commutative integration has its origin in the classical papers of Murray and von Neumann on rings of operators, and was introduced because of unsolved problems in unitary group representations and the elucidation of various aspects of quantum-mechanic
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MATHEMA TICAL PHYSICS STUDIES A SUPPLEM ENT AR Y SER IES TO LETTERS IN MATHEMATICAL PHYSICS
Editors: J. C. CORTET, UniversitedeDijon, France M. FLA TO, Universite de Dijon, France M. GUENIN, Institut de Physique Theorique, Geneva, Switzerland E. H. LIEB, Princeton University, US.A. R. RACZKA, Institute of Nuclear Research, Warsaw, Poland
Editorial Board:
w. AM REIN, Institut de Physique Theorique, Geneva, Switzerland H. ARAKI, Kyoto University, Japan A. CONNES, I.H.E.S., France L. FAD D E EV , Steklov Institute of Mathematics, Leningrad, U.S.S.R. J. FROHLICH, F.T.H., Switzerland C. FRONSDAL, UCLA, Los Angeles, U.S.A. I. M. GELFAND, Moscow State University, US.S.R. A. JAFFE, Harvard University, U.S.A. A. A. KIRILLOV, Moscow State University, U.S.S.R. A. LICHNEROWICZ, College de France, France B. NAGEL, K.T.H., Stockholm, Sweden J. NIEDERLE, Institute of Physics CSAV, Prague, Czechoslovakia A. SALAM, International Center for Theoretical Physics, Trieste, Italy W. SCHMID, Harvard University, U.S.A.
I.E.SEGAL,M.I.T., US.A. J. SIM 0 N, Universitye de Dijon, France D. STERNHEIMER, Col/ege de France, France I. T. TODOROV, Institute of Nuclear Research, Sofia, Bulgaria
VOLUME 7
Unbounded Non-Commutative Integration by
J. P. Jurzak Physique-Mathematique, Universiu! de Dijon, France
D. Reidel Publishing Company A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
Dordrecht / Boston / Lancaster
Library of Congrets Ca taloging in Publication Data JUlZak., J . P. (Jean· Paul ). 1950-
Unbou nded non·commutative inlcg/ation. (Mathematical physics studies: v. 7) Bibliography: p. Includes index. \. Integration. Functional. 2. Von Neumann algebras. 3. Noncommutative algebras.. Ope rator theory. 5. Mathematical physics. I. Title. II. Series. QC20. 7. r85J 87 1985 530.)'5 15 7 85-10759 15BN· 13: 978·94·010·88t3· 8 e·ISBN· 13: 978·94·009·5231 · 7 001: 10.10071978-94·009· 5231 ·7
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All Righ ts Reserved Cl 1985 by D. Reidel Publishing Comp any, Dordrccht, Holland Softcover reprint of the hardcover 1st editio n 1985 No pall of the material protected by this copyright notice may be Icproduoed or utilized in any fOlm o r by any means, electronic or mechanical, including photocopying, rccording o r by any information storage and retrieval system, without written pelmission from the copy right owner
A DOMINIQUE
TABLE OF CONTENTS
PREFACE
ix
INTRODUCTION
xi
VOCABULARY
xvii
EXAMPLES AND OBSERVATIONS CHAPTER 1: MAIN SPACES
7
CHAPTER 2: DENSITY THEOREMS
21
CHAPTER 3: TECHNICAL PROPERTIES OF THE DOMAIN
30
CHAPTER 4: ELEMENTARY OPERATIONS
57
CHAPTER 5: GELFAND TRANSFORMATION
65
CHAPTER 6: COFINAL CENTRAL SYSTEMS AND DERIVATIONS
83
CHAPTER 7: NOTION OF REPRESENTATION: THE G.N.S. C
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