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Uniform Algebras

Theodore W. Gamelin
Publisher: 
Amaerican Mathematical Society/Chelsea
Publication Date: 
1984
Number of Pages: 
269
Format: 
Hardcover
Edition: 
2
Price: 
39.00
ISBN: 
0828403112
Category: 
Monograph
BLL Rating: 

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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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