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D. Van Nostrand Company, Inc. Toronto New York London New York Introduction to Abstract Harmonic Analysis by LYNN H. LOOMIS Associate Professor of Mathematics Harvard University 1953 D. VAN NOSTRAND COMPANY, INC. TORONTO NEW YORK LONDON NEW YORK D. Van Nostrand Company, Inc., 250 Fourth Avenue, New York 3 TORONTO D. Van Nostrand Company (Canada), Ltd., 25 Hollinger Rd., Toronto LONDON Macmillan & Company, Ltd., St. Martin's Street, London, W.C. 2 COPYRIGHT, 1953, BV D. VAN NOSTRAND COMPANY, INC. Published simultaneously in Canada by D. VAN NOSTRAND COMPANY (Canada), LTD. All Rights Reserved This book, or any parts thereof, may not be reproduced in any form without written per- mission from the author and the publisher. Library of Congress Catalogue Card No. 53-5459 PRINTED IN THE UNITED STATES OF AMERICA PREFACE This book is an outcome of a course given at Harvard first by G. W. Mackey and later by the author. The original course was modeled on Weil's book [48] and covered essentially the material of that book with modifications. As Gelfand's theory of Banach algebras and its applicability to harmonic analysis on groups became better k lown, the methods and content of the course in- evitably shifted in this direction, and the present volume concerns itself almost exclusively with this point of view. Thus our devel- opment of the subject centers around Chapters IV and V, in which the elementary theory of Banach algebras is worked out, and groups are relegated to the supporting role of being the prin- cipal application. A typical result of this shift in emphasis and method is our treatment of the Plancherel theorem. This theorem was first formulated and proved as a theorem on a general locally compact Abelian group by Weil. His proof involved the structure theory of groups and was difficult. Then Krein [29] discovered, apparently without knowledge of Weil's theorem, that the Plancherel theorem was a natural consequence of the application of Gelfand's theory to l the L algebra of the group. This led quite naturally to the form- ulation of the theorem in the setting of an abstract commutative (Banach) algebra with an involution, and we follow Godement [20] in taking this as our basic Plancherel theorem. (An alternate proof of the Plancherel theorem on groups, based on the Krein- Milman theorem, was given by Cartan and Godement [8].) The choice of Banach algebras as principal theme implies the neglect of certain other tools which are nevertheless important in the investigation of general harmonic analysis, such as the Krein- Milman theorem and von Neumann's direct integral theory, each of which is susceptible of systematic application. It is believed, however, that the elementary theory of Banach algebras and its vi PREFACE applications constitute a portion of the subject which in itself is of general interest and usefulness, which can serve as an introduc- tion to present-day research in the whole field, and whose treat- ment has already acquired an elegance promising some measure of permanence. The specific prerequisites for the reader include a knowledge of the concepts of elementary modern algebra and of metric space topology. In addition, Liouville's theorem from the elementary theory of analytic functions is used twice. In practice it will probably be found that the requirements go further than this, for without some acquaintance with measure theory, general topology, and Banach space theory the reader may find the pre- liminary material to be such heavy going as to overshadow the main portions of the book. This preliminary material, in Chapters I-III, is therefore presented in a condensed form under the as- sumption that the reader will not be unfamiliar with the ideas being discussed. Moreover, the topics treated are limited to those which are necessary for later chapters; no effort has been made to write small textbooks of these subjects. With these restrictions in mind, it is nevertheless hoped that the reader will find these chapters self-contained and sufficient for all later purposes. As we have mentioned above, Chapter IV is the central chapter of the book, containing an exposition of the elements of the theory of Banach algebras in, it is hoped, a more leisurely and systematic manner than found in the first three chapters. Chapter V treats certain special Banach algebras and introduces some of the notions and theorems of harmonic analysis in their abstract forms. Chap- ter VI is devoted to proving the existence and uniqueness of the Haar integral on an arbitrary locally compact group, and in Chap- ters VII and VIII the theory of Banach algebras is applied to deduce the standard theory of harmonic analysis on locally com- pact Abelian groups and compact groups respectively. Topics covered in Chapter VII include positive definite functions and the generalized Bochner theorem, the Fourier transform and Plancherel theorem, the Wiener Tauberian theorem and the Pontriagin duality theorem. Chapter VIII is concerned with PREFACE vii representation theory and the theory of almost periodic functions. We conclude in Chapter IX with a few pages of introduction to the problems and literature of some of the areas in which results are incomplete and interest remains high. CONTENTS CHAPTER I: TOPOLOGY SECTION PAGE 1. Sets 1 2. Topology 3 3. Separation axioms and theorems 5 4. The Stone-Weierstrass theorem 8 5. Cartesian products and weak topology 10 CHAPTER II: BANACH SPACES 6. Normed linear spaces 13 7. Bounded linear transformations 15 8. Linear functionals '. 18 9. The weak topology for A"* 22 10. Hilbert space 23 11. Involution on (B(/7) 27 CHAPTER III: INTEGRATION 12. The Daniell integral 29 13. Equivalence and measurability 34 p 14. The real L -spaces 37 15. The conjugate space of IT 40 16. Integration on locally compact Hausdorff spaces 43 17. The complex //-spaces 46 CHAPTER IV. BANACH ALGEBRAS 18. Definition and examples 48 19. Function algebras 50 20. Maximal ideals 58 21. Spectrum; adverse 63 22. Banach algebras; elementary theory 66 23. The maximal ideal space of a commutative Banach algebra . 69 24. Some basic general theorems 75 ix CONTENTS CHAPTER V: SOME SPECIAL BANACH ALGEBRAS SECTION PAGE 25. Regular commutative Banach algebras 82 26. Banach algebras with involutions 87 27. #*-algebras 100 CHAPTER VI: THE HAAR INTEGRAL 28. The topology of locally compact groups 108 29. The Haar integral 112 30. The modular function 117 31. The group algebra 120 32. Representations 127 33. Quotient measures 130 CHAPTER VII : LOCALLY COMPACT ABELIAN GROUPS 34. The character group 134 35. Examples 138 36. The Bochner and Plancherel theorems 141 37. Miscellaneous theorems 147 38. Compact Abelian groups and generalized Fourier series . 153 CHAPTER VIII : COMPACT GROUPS AND ALMOST PERIODIC FUNCTIONS 39. The group algebra of a compact group 156 40. Representation theory 161 41. Almost periodic functions 165 CHAPTER IX : SOME FURTHER DEVELOPMENTS 42. Non-commutative theory 174 43. Commutative theory 179 Bibliography 185 Index 188 Chapter I TOPOLOGY 1. SETS 1A. The reader is assumed to be familiar with the algebra of sets, and the present paragraph is confined to notation. The curly bracket notation is used to name specific sets. Thus {a, b} is set elements are a and and a is the set the whose b, {a ly , n } a . sets whose elements are a\ y , n However, cannot generally be named by listing their elements, and more often are defined by properties, the notation being that {x: ( )} is the set of all x such that ( ). We write "a G A" for "a is a member of A" = and "5 c A" for "5 is a subset of ^"; thus A {x: x C A\. The ordinary "cup" and "cap" notation will be used for com- binations of sets: A U B for the union of A and By and A D B for their intersection; Un=i An for the union of the sets of the countable and for their intersection. family [An } y fln=i An More generally, if F is any family of sets, then U^eff ^ or is the union of the sets in and U \A\ A G $} ff, {\{A\ A _$} of is is their intersection. The complement Ay designated A' y understood to be taken with respect to the "space" in question, that is, with respect to whatever class is the domain of the given discussion. The null class is denoted by 0. The only symbols from logic which will be used are "=" for "if then" and "=>" for "if and only if." IB. A binary relation "<" between elements of a class A is called a partial ordering of A (in the weak sense) if it is transitive = for a (a < b and b < c a < c) y reflexive (a < a every G A) and if a < b and b < a => a = b. A v$> called a directed set l TOPOLOGY " " under <" if it is partially ordered by <" and if for every a and b G A there exists c G A such that c < a and c < b. More properly, A in this case should be said to be directed downward; there is an obvious dual definition for being directed upward. A partially ordered set A is linearly ordered if either a < b or b < a for every distinct pair a and b C, A. A is partially ordered in the strong sense by "<" if "<" \ s transitive and irreflexive (a < a). In this case "gs" is evidently a corresponding weak partial ordering, and, conversely, every weak partial ordering has an associated strong partial ordering. 1C. The fundamental axiom of set theory which has equiva- lent formulations in Zorn's lemma, the axiom of choice and the well-ordering hypothesis will be freely used throughout this book.
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