AMS Chelsea Publishing 1984; 269 pp; hardcover Volume: 311 ISBN10: 0821840495 ISBN13: 9780821840498 List Price: US$46 Individual Members: US$41.40 Order Code: CHEL/311.H
 From the Preface: "The functionalanalytic approach to uniform algebras is inextricably interwoven with the theory of analytic functions ... [T]he concepts and techniques introduced to deal with these problems [of uniform algebras], such as "peak points" and "parts," provide new insights into the classical theory of approximation by analytic functions. In some cases, elegant proofs of old results are obtained by abstract methods. The new concepts also lead to new problems in classical function theory, which serve to enliven and refresh that subject. In short, the relation between functional analysis and the analytic theory is both fascinating and complex, and it serves to enrich and deepen each of the respective disciplines." This volume includes a Bibliography, List of Special Symbols, and an Index. Each of the chapters is followed by notes and numerous exercises. Readership Graduate students and research mathematicians interested in analysis. Reviews "This text is a classical reference on uniform algebras, written by a major expert in the area. ... Gamelin's book is a nice text that any mathematician interested in uniform algebras or related questions should have on his shelves. The book is also interesting for nonspecialists. ... These results follow from the beautiful abstract theory developed in the first two chapters of the book, and their proofs differ from the ones in the usual text books on analytic functions. The mathematical exposition in the book is excellent."  Journal of Approximation Theory Table of Contents Commutative Banach Algebras  1. Spectrum and resolvent
 2. The maximal ideal space
 3. Examples
 4. The Shilov boundary
 5. Two basic theorems
 6. Hulls and kernels
 7. Commutative \(B^\ast\)algebras
 8. Compactifications
 9. The algebra \(L^{\infty}\)
 10. Normal operators on Hilbert space
 Notes
 Exercises
Uniform Algebras  1. Algebras on planar sets
 2. Representing measures
 3. Dirichlet algebras
 4. Logmodular algebras
 5. Maximal subalgebras
 6. Hulls
 7. Decomposition of orthogonal measures
 8. Cauchy transform
 9. Mergelyan's theorem
 10. Local algebras
 11. Peak points
 12. Peak sets
 13. Antisymmetric algebras
 Notes
 Exercises
Methods of Several Complex Variables  1. Polynomial convexity
 2. Rational convexity
 3. Circled sets
 4. Functional calculus
 5. Polynomial approximation
 6. Implicit function theorem
 7. Cohomology of the maximal ideal space
 8. Local maximum modulus principle
 9. Extensions of uniform algebras
 Notes
 Exercises
Hardy Spaces  1. The conjugation operator
 2. Representing measures for \(H^{\infty}\)
 3. The uniqueness subspace
 4. Enveloped measures
 5. Core measures
 6. The finite dimensional case
 7. Logmodular measures
 8. Hypodirichlet algebras
 Notes
 Exercises
Invariant Subspace Theory  1. Uniform integrability
 2. The Hardy algebra
 3. Jensen measures
 4. Characterization of \(H\)
 5. Invertible elements of \(H\)
 6. Invariant subspaces
 7. Embedding of analytic discs
 8. Szegö's theorem
 9. Extremal functions in \(H^1\)
 Notes
 Exercises
Parts  1. Representing measures for a part
 2. Characterization of parts
 3. Parts of \(R(K)\)
 4. Finitely connected case
 5. Pointwise bounded approximation
 6. Finitely generated ideals
 7. Extremal methods
 Notes
 Exercises
Generalized Analytic Functions  1. Preliminaries
 2. Algebras associated with groups
 3. A theorem of Bochner
 4. Generalized analytic functions
 5. Analytic measures
 6. Local product decomposition
 7. The Hardy spaces
 8. Weakstar maximality
 9. Weight functions
 10. Invariant subspaces
 11. Structure of cocycles
 12. Cocycles and invariant subspaces
 Notes
 Exercises
Analytic Capacity and Rational Approximation  1. Analytic capacity
 2. Elements of analytic capacity
 3. Continuous analytic capacity
 4. Peaking criteria
 5. Criteria for \(R(K)=C(K)\)
 6. Analytic diameter
 7. A scheme for approximation
 8. Criteria for \(R(K)=A(K)\)
 9. Failure of approximation
 10. Pointwise bounded approximation
 11. Pointwise bounded approximation with same norm
 12. Estimates for integrals
 13. Analytically negligible sets
 Notes
 Exercises
 Bibliography
 List of special symbols
 Index
