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 b