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Uniform Algebras: Second Edition
Theodore W. Gamelin
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AMS Chelsea Publishing
1984; 269 pp; hardcover
Volume: 311
ISBN-10: 0-8218-4049-5
ISBN-13: 978-0-8218-4049-8
List Price: US$43
Individual Members: US$38.70
Order Code: CHEL/311.H
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From the Preface: "The functional-analytic 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

"The Mathematical exposition in the book is excellent, and at the end of each chapter there is a short section which explains the historic development of the topics studied in the chapter."

-- 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. Weak-star 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
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