Introduction to Banach Spaces and Algebras

380 Pages·2011·40.96 MB·English

by  Graham R. Allan

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20. Graham R. Allan: Introduction to Banach Spaces and Algebras
Introduction to Banach
Spaces and Algebras
as S! ;22
Graham R. Allan
University of Cambridge
Prepared for publication by
H. Garth Dales
University of Leeds
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Preface
This book is based on lectures that were given over several years by Dr. Graham
Allan, my supervisor, teacher and friend, to 'Part III' students of mathematics
at Cambridge University.
Graham had produced notes for the students as background to the lectures,
and he was invited by Oxford University Press to consolidate these into a book
on the topic of the lectures. Indeed, he made considerable progress on this, and
had created a 1EX file of material forming the backbone of the present work.
Sadly, he became seriously ill in autumn 2006, and was not able to finish this
task; in 2007, he and I discussed the project, and we agreed that I should take
responsibility for completing the preparation of the work for final publication.
Graham indicated in broad terms the topics that were not yet in the notes,
and that he hoped would be included. Thus the original material has been ex
panded to include these specific topics, and also various related topics that I felt
were natural to the subject and in the spirit of his intentions. Further, the text
as it stood when it came to me did not contain any exercises, although some
of these were available from his other files. Thus the general structure of the
book, organization of the sections, and much of the material is Graham's; I take
responsibility for some details of the presentation, for the additional material,
and, of course, for any errors that remain. Further, all the notes and exercises
at the end of sections, and also the bibliography, are due to myself. In total, the
text has increased by about 40% in length from Graham's original files.
Graham Allan died in August 2007, and was not able to see my version of
his text. An obituary, including an appreciation of his mathematical work, will
appear in the Bulletin of the London Mathematical Society.
I attended Graham's Part III lectures on Banach algebras in the year 1966-
67. I found the subject to be an exciting blend of algebra and analysis, and was
attracted by the clarity and beauty of his exposition; I then became his graduate
stUdent, and have continued to work on Banach algebras and related topics for
the remainder of my career in mathematics.
Throughout the text, I have tried to maintain the tone and style of Graham's
notes. Reading the text, I hear his clear and modest voice speaking poignantly
from the page; I hope and trust that he would have approved of the final mani
festation of his mathematical thoughts and intentions.
It is evident that Graham's lectures inspired many students who attended his
Part III lectures to continue in the subject, and many became his own graduate
students over the years. These graduate students include the following (with the
year that their PhD was awarded): John F. Rennison (1968), Ian G. Craw (1970),
H. Garth Dales (1970), J. Peter McClure (1970), Peter G. Dixon (1970), Ghod
sieh Vakily (1974), Y. Nejad-Degan (1977), Amir Khosravi (1981), Thomas J.
Ransford (1984), Frederic Gourdeau (1989), Michael C. White (1990), Christo
pher E. J. Kilgour (1994), Simon E. Morris (1994), Thomas Vils Pedersen (1994),
Timothy J. D. Wilkins (1996), Michael K. Kopp (2003), and Daniel L. Neale
(2005).
vi Preface
As students of Graham, we would like to express our admiration, affection,
and respect for his fine mathematics and his personal kindness and integrity, and
our deep sense of loss at his passing.
I am very grateful to friends who have read some or all of the manuscript at
various stages, and who made suggestions and indicated errors. These include
Joel Feinstein (Nottingham), E. Christopher Lance (Leeds), Richard Loy (Can
berra), Shital Patel (Kanpur), Hung Le Pham (Wellington), Thomas Ransford
(Quebec), David Salinger (Leeds), and Dona Strauss (Leeds).
I am also grateful to Graham's widow, Mrs. Elizabeth Allan, for her encour
agement and support in this project.
H. G. Dales, Leeds, July, 2010.
Contents
Introduction 1
PART I INTRODUCTION TO BANACH SPACES
1. Preliminaries 7
Remarks on set theory 8
Metric spaces and analytic topology 16
Complex analysis 30
2. Elements of normed spaces 33
Definitions and basic examples 33
Weierstrass approximation theorems 60
Inner-product spaces 70
Elementary ideas on Fourier series 84
Fourier integrals and Hermite functions 98
3. Banach spaces 106
Existence of continuous linear functionals 106
Separation theorems 122
Category theorems 128
Dual operators 140
PART II INTRODUCTION TO BANACH ALGEBRAS
4. Banach algebras 155
Elementary theory 155
Commutative Banach algebras 185
Runge's theorem and the holomorphic functional calculus 211
5. Representation theory 226
Representations and modules 226
Automatic continuity 241
Variation of the spectral radius 248
6. Algebras with an involution 260
Banach algebras with an involution 260
C* -algebras 269
7. The Borel functional calculus 285
The Daniell integral 285
The Borel functional calculus and the spectral theorem 294
viii Contents
PART III SEVERAL COMPLEX VARIABLES AND BANACH
ALGEBRAS
8. Introduction to several complex variables 305
Differentiable functions in the plane 305
Functions of several variables 313
Polynomial convexity 326
9. The holomorphic functional calculus in several variables 339
The main theorem 339
Applications of the functional calculus 345
References 354
Index of terms 363
Index of symbols 369
Introduction
This book is based on lectures that were given to 'Part III' students at Cambridge
over several years; Part III of the Mathematical Tripos at Cambridge is usually
taken by students in their fourth year of studies of mathematics at the University.
Thus the book is suitable for strong students in their final year of a first degree at
a university, for students taking a Master's degree or a 'second cycle' qualification
in pure mathematics, and for graduate students of pure mathematics in their
earlier years. We hope that the book will also be valuable for those in other
mathematical disciplines who seek an introduction to functional analysis and
Banach algebra theory as a background to their own studies. The book represents
an extended version of the union of several courses that were given at Cambridge
at different times.
As a background, we are assuming knowledge of material usually found in the
first three years of a degree in mathematics. These preliminaries are summarized
in Chapter 1: basically, in the text, we shall assume knowledge of usual 'naive'
set theory, of the theory of metric spaces and more general topological spaces
(but not including general theories of convergence involving nets), and elemen
tary complex analysis in one variable. We shall also assume knowledge of linear
algebra and the theory of associative algebras, up to the elementary theory of
ideals.
In the notes at the end of sections, and occasionally in the exercises, we do
assume somewhat wider background knowledge. Indeed, the notes and exercises
sketch further developments of the theory that was given in the text, and indicate
sources where our subject is taken further.
We have made a conscious decision not to assume knowledge of measure the
ory and Lebesgue integration; this topic is often covered in a parallel or later
course at Cambridge. We do assume knowledge of undergraduate Riemann inte
gration theory; a few comments in the notes and the exercises are more naturally
cast in the language of Lebesgue theory, and will be appreciated by those who
have studied this subject. In a similar way, some calculations of cardinalities in
the exercises assumed rather more knowledge of set theory than is required in
the main text.
The subject of Banach and Hilbert spaces, and of linear operators on these
spaces, has a history of more than one hundred years. Near the beginning of
the twentieth century, mathematicians were led to develop an abstract setting
in which many classical problems could be expressed; these problems included
those of trigonometric series, of the spectral theory of 'infinite matrices', and
the extension of geometrical ideas involving finite-dimensional Euclidean space
2 Introduction
to 'infinite-dimensional Hilbert spaces'. For example, to capture the idea of the
frequencies of vibrating strings, one needs 'infinitely many degrees of freedom'.
Great principles of functional analysis, such as the Hahn-Banach theorem and
the principle of uniform boundedness, appeared in the 1920s, and in 1932 Banach
crystallized what are now called 'Banach spaces' in his great monograph of that
year. A vast corpus of theory and applications of this subject has been developed
over time, and it seems certain that the language of Banach spaces is ideally
suited to capturing the abstract essence of many classical and modern topics in
mathematics. The main topics of the 'post-Banach period' that we shall expound
are those of weak and weak-* topologies, the Krein-Milman theorem on extreme
points, and the beginnings of an approach to 'abstract harmonic analysis' and
the theory of Fourier transforms.
There is an immense body of work on Banach spaces, on operators between
these spaces, and on their applications. For example, the classic texts of Dunford
and Schwartz, starting in 1958, already have more than 2500 pages on the single
subject of 'linear operators'. Thus our work can be only an introduction to these
subjects.
The theory of Banach algebras grew out of that of Banach spaces: it adds
to the 'infinite-dimensional linear analysis' of Banach-space theory the concept
of an algebra, with a product. The subject begins with the work of Gel'fand
around 1940, and so has been studied for at least 70 years. Our claim about the
subject is, first, that Banach algebra theory captures the essence of many key
classical examples: commutative Banach algebras of functions illuminate many
questions of approximation theory; abstract harmonic analysis, based on the
group algebra of a locally compact group, is the perfect setting for studies of
Fourier transforms; non-commutative algebras of bounded linear operators on
Banach and Hilbert spaces are the natural and necessary generalizations of the
notion of a matrix acting on a Euclidean space, with their profound applications
to mathematical physics. Our second claim is that in Banach algebra theory there
is a beautiful and subtle blend of ideas from analysis and algebra. A key result
in this blend is Johnson's uniqueness-of-norm theorem, which shows that the
apparently weak condition joining the Banach space and algebraic structures
by the requirement that the multiplication be continuous already ensures the
deep connection that any two norms making a semisimple algebra into a Banach
algebra must determine the same topology on the algebra.
As well as being solidly based in algebra, Banach algebra theory is deeply
entwined with the theory of complex analysis; first with the theory of analytic
functions on open sets in the plane C, and then, for deeper results, with analytic
functions of several complex variables. The former theory is generally taught
at undergraduate level and we shall assume that it is known; the latter is not
usually so taught, and so we shall develop the background results in this subject
that we shall require.
The text is divided into three parts, each of several chapters. Each of the three
parts was the basis of one lecture course, but the material has been extended