Uniform Algebras

A classical problem in analysis is to find conditions under which a given complex-valued function on a topological space can be uniformly approximated by elements of a prescribed algebra of functions on this set. It turns out that an important role in solving this problem is played by uniform algebras, which constitute an important and interesting class of commutative Banach algebras. The present book is devoted to the detailed study of uniform algebras. In addition to using methods and results of the theory of Banach algebras (the structure of the space of maximal ideals), the theory of uniform algebras is related to the theory of analytic functions, so that some of the deepest results about uniform algebras involve applying the theory of analytic functions in several variables. A significant part of the book is devoted to this important interplay between uniform algebras and analytic function theory. The prerequisites for reading this volume are real analysis and basic notions on the theory of Banach spaces.