During the past twenty years many connections have been found between thetheory of analytic functions of one or more complex variables and the study ofcommutative Banach algebras. On the one hand, function theory has been used toanswer algebraic questions such as the question of the existence of idempotents ina Banach algebra. On the other hand, concepts arising from the study of Banachalgebras 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.Wepresupposenoknowledgeofseveralcomplexvariablesonthepartofthereaderbut develop the necessary material from scratch. The remainder of the book dealswith problems of uniform approximation on compact subsets of the space of ncomplex variables. For n &gt; 1 no complete theory exists but many importantparticular 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 alongthe 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 treatedin 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 includingmany examples is given by E. L. Stout, The Theory of Uniform Algebras, (Bogdenand Quigley, Inc., 1971).The first edition of this book was published in 1971 by Markham PublishingCompany. 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 readershould try to solve as many of these as he can. Solutions to starred exercises aregiven in Section 22.

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