Banach Algebras and Several Complex Variables

John Wermer (auth.)0387982531, 9780387982533, 9780387225869

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.

---

Table of contents :
Front Matter….Pages i-ix
Preliminaries and Notations….Pages 1-4
Classical Approximation Theorems….Pages 5-16
Operational Calculus in One Variable….Pages 17-22
Differential Forms….Pages 23-26
The Oka—Weil Theorem….Pages 27-30
Operational Calculus in Several Variables….Pages 31-35
The Šilov Boundary….Pages 36-42
Maximality and Radó’s Theorem….Pages 43-49
Analytic Structure….Pages 50-56
Algebras of Analytic Functions….Pages 57-63
Approximation on Curves in C n ….Pages 64-70
Uniform Approximation on Disks in C n ….Pages 71-76
The First Cohomology Group of a Maximal Ideal Space….Pages 77-81
Manifolds Without Complex Tangents….Pages 82-86
Submanifolds of High Dimension….Pages 87-95
Generators….Pages 96-110
The Fibers Over a Plane Domain….Pages 111-121
Examples of Hulls….Pages 122-130
Solutions to Some Exercises….Pages 131-136
Back Matter….Pages 137-142
….Pages 143-149