Table Of ContentAn Introduction to
Abstract Harmonic Analysis
by
LYNN H. LOOMIS
Associate Professor of Mathematics
Harvard University
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Copyright© 1953, by D. VAN NOSTRAND COMPANY, INC.
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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 known, 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 ofWeil's theorem, that the Plancherel theorem
was a natutal consequence of the application of Gelfand's theory to
the V 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
y
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 Vll
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. N armed linear spaces . . . . 13
7. Bounded linear transformations 15
8. Linear functionals . . . . 18
9. The weak topology for X* 22
10. Hilbert space . . . 23
11. Involution on ffi(H) 27
CHAPTER III: INTEGRATION
12. The Daniell integral . . . . . 29
13. Equivalence and measurability 34
14. The real LP-spaces . . . . . . 37
15. The conjugate space of LP 40
16. Integration on locally compact Hausdorff spaces . 43
17. The complex LP-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
X CONTENTS
CHAPTER V: SOME SPECIAL BANACH ALGEBRAS
SECTION PAGE
25. Regular commutative Banach algebras 82
26. Banach algebras with involutions 87
27. H*-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 . . . . . . . 14 7
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
lA. 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 the set whose elements are a and b, and {a h · · ·, an} is the set
whose elements are ah · · ·, an. However, sets 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 E: A" for "a is a member of A"
and" B c A" for" B is a subset of A"; thus A= {x: x E: A}.
The ordinary "cup" and "cap" notation will be used for com
binations of sets: A U B for the union of A and B, and A n B
U.7=
for their intersection; An for the union of the sets of the
1 n:=t
countable family {An}, and An for their intersection.
UAO
More generally, if ff' is any family of sets, thenn A or
U
{A: A E: ff'} is the union of the sets in ff', and {A: A E: ff'}
is their intersection. The complement of A, designated A', is
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."
lB. A binary relation "<" between elements of a class A is
called a partial ordering of A (in the weak sense) if it is transitive
(a < b and b < c ~ a < c), reflexive (a < a for every a E: A)
and if a < b and b < a ~ a = b. A is called a directed set
2 TOPOLOGY
under " <" if it is partially ordered by " <" and if for every a
and b E:: A there exists c E::: 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 E::: A. A is partially ordered
in the strong sense by " <" if " <" is transitive and irrejlexiue
(a <( a). In this case " ~" is evidently a corresponding weak
partial ordering, and, conversely, every weak partial ordering has
an associated strong partial ordering.
lC. 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.
In fact, its use is absolutely essential for the success of certain
abstract methods such as found in Gelfand's theory of Banach
algebras. Many mathematicians feel dubious about the validity
of the' axiom of choice, but it must be remembered that Godel
has shown that, if mathematics is consistent without the axiom
of choice, then it remains consistent if this axiom is added (and
similarly for the generalized continuum hypothesis). Thus a
theorem which is proved with the aid of the axiom of choice can
never be disproved, unless mathematics already contains an in
consistency which has nothing to do with the axiom of choice.
Zorn's lemma. Euery partially ordered set A includes a maxi
ma/linearly ordered subset. If euery linearly ordered subset of A
has an upper bound in A, then A contains a maximum element.
The second statement follows from the first upon taking an
upper bound x of a maximal linearly ordered subset B. Then x
is a maximal element of A, for any properly larger element could
be added to B without destroying its linear order, contradicting
the maximal character of B. We sketch below a proof of the
axiom of choice from Zorn's lemma. Of these two properties, it
is a matter of mere preference which is taken as an axiom and
which is proved as a theorem. However, the proof given here
is much easier than the converse proof.
lD. Theorem (Axiom of choice). If F is a function with do
main D such that F(x) is a non-empty set for euery x E::: D, then
