Several Complex Variables and Banach Algebras PDF
Preview Several Complex Variables and Banach Algebras
During the past twenty years many connections have been found between the
theory of analytic functions of one or more complex variables and the study of
commutative Banach algebras. On the one hand, function theory has been used to
answer algebraic questions such as the question of the existence of idempotents in
a Banach algebra. On the other hand, concepts arising from the study of Banach
algebras such as the maximal ideal space, the ˇ Silov boundary, Gleason parts, etc.
have led to new questions and to new methods of proof in function theory.
Roughlyonethirdofthisbookisconcernedwithdevelopingsomeoftheprinci-
palapplicationsoffunctiontheoryinseveralcomplexvariablestoBanachalgebras.
Wepresupposenoknowledgeofseveralcomplexvariablesonthepartofthereader
but develop the necessary material from scratch. The remainder of the book deals
with problems of uniform approximation on compact subsets of the space of n
complex variables. For n > 1 no complete theory exists but many important
particular problems have been solved.
Throughout, our aim has been to make the exposition elementary and self-
contained. We have cheerfully sacrificed generality and completeness all along
the way in order to make it easier to understand the main ideas.
Relationships between function theory in the complex plane and Banach alge-
bras are only touched on in this book. This subject matter is thoroughly treated
in A. Browder’s Introduction to Function Algebras, (W. A. Benjamin, New York,
1969) and T. W. Gamelin’s Uniform Algebras, (Prentice-Hall, Englewood Cliffs,
N.J., 1969). A systematic exposition of the subject of uniform algebras including
many examples is given by E. L. Stout, The Theory of Uniform Algebras, (Bogden
and Quigley, Inc., 1971).
The first edition of this book was published in 1971 by Markham Publishing
Company. The present edition contains the following new Sections: 18. Subman-
ifolds of High Dimension, 19. Generators, 20. The Fibers Over a Plane Domain,
21. Examples of Hulls. Also, Section 11 has been revised.
Exercises of varying degrees of difficulty are included in the text and the reader
should try to solve as many of these as he can. Solutions to starred exercises are
given in Section 22.
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