Ebook: Several Complex Variables and Banach Algebras

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.

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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.

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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.
Content:
Front Matter....Pages i-xii
Preliminaries and Notation....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 ?ilov Boundary....Pages 36-42
Maximality and Rad?’s Theorem....Pages 43-49
Maximum Modulus Algebras....Pages 50-56
Hulls of Curves and Arcs....Pages 57-63
Integral Kernels....Pages 64-83
Perturbations of the Stone?Weierstrass Theorem....Pages 84-91
The First Cohomology Group of a Maximal Ideal Space....Pages 92-101
Manifolds Without Complex Tangents....Pages 102-111
Submanifolds of High Dimension....Pages 112-119
Boundaries of Analytic Varieties....Pages 120-133
Polynomial Hulls of Sets Over the Circle....Pages 134-145
Areas....Pages 146-154
Topology of Hulls....Pages 155-169
Examples....Pages 170-179
Historical Comments and Recent Developments....Pages 180-186
Appendix....Pages 187-193
Solutions to Some Exercises....Pages 194-205
Back Matter....Pages 206-223
....Pages 224-230