Preliminaries and Notation.- Classical Approximation Theorems.- Operational Calculus in One Variable.- Differential Forms.- The -Operator.- The Equation .- The Oka-Weil Theorem.- Operational Calculus in Several Variables.- The Šilov Boundary.- Maximality and Radó's Theorem.- Maximum Modulus Algebras.- Hulls of Curves and Arcs.- Integral Kernels.- Perturbations of the Stone?Weierstrass Theorem.- The First Cohomology Group of a Maximal Ideal Space.- The -Operator in Smoothly Bounded Domains.- Manifolds Without Complex Tangents.- Submanifolds of High Dimension.- Boundaries of Analytic Varieties.- Polynomial Hulls of Sets Over the Circle.- Areas.- Topology of Hulls.- Pseudoconvex sets in ?n.- Examples.- Historical Comments and Recent Developments.- Solutions to Some Exercises.

Many connections have been found between the theory of analytic functions of one or more complex variables and the study of commutative Banach algebras. While function theory has often been employed to answer algebraic questions such as the existence of idempotents in a Banach algebra, concepts arising from the study of Banach algebras including the maximal ideal space, the Silov boundary, Geason parts, etc. have led to new questions and to new methods of proofs in function theory. This book is concerned with developing some of the principal applications of function theory in several complex variables to Banach algebras. The authors do not presuppose any knowledge of several complex variables on the part of the reader and all relevant material is developed within the text. Furthermore, the book deals with problems of uniform approximation on compact subsets of the space of n complex variables. The third edition of this book contains new material on; maximum modulus algebras and subharmonicity, the hull of a smooth curve, integral kernels, perturbations of the Stone-Weierstrass Theorem, boundaries of analytic varieties, polynomial hulls of sets over the circle, areas, and the topology of hulls. The authors have also included a new chapter containing commentaries on history and recent developments and an updated and expanded reading list.