Uniform Algebras

by Gamelin, Theodore W.

• ISBN13: 9780821840498
• ISBN10: 0821840495
• Format: Hardcover
• Copyright: 2005-06-01
• Publisher: Amer Mathematical Society

Summary

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.

Table of Contents

Commutative Banach Algebras

Spectrum and resolvent

The maximal ideal space

Examples

The Shilov boundary

Two basic theorems

Hulls and kernels

Commutative $B^\ast$-algebras

Compactifications

The algebra $L^{\infty}$

Normal operators on Hilbert space Note

Exercises Uniform Algebras

Algebras on planar sets

Representing measures

Dirichlet algebras

Logmodular algebras

Maximal subalgebras

Hulls

Decomposition of orthogonal measures

Cauchy transform

Mergelyan's theorem

Local algebras

Peak points

Peak sets

Antisymmetric algebras Notes Exercises

Methods of Several Complex Variables

Polynomial convexity

Rational convexity

Circled sets

Functional calculus

Polynomial approximation

Implicit function theorem

Cohomology of the maximal ideal space

Local maximum modulus principle

Extensions of uniform algebras Notes Exercises

Hardy Spaces

The conjugation operator

Representing measures for $H^{\infty}$

The uniqueness subspace

Enveloped measures

Core measures

The finite dimensional case

Logmodular measures

Hypodirichlet algebras Notes Exercises

Invariant Subspace Theory

Uniform integrability

The Hardy algebra

Jensen measures

Characterization of $H$

Invertible elements of $H$

Invariant subspaces

Embedding of analytic discs

Szego's theorem

Extremal functions in $H^1$ Notes Exercises

Parts

Representing measures for a Part

Characterization of parts

Parts of $R(K)$

Finitely connected case

Pointwise bounded approximation

Finitely generated ideals

Extremal methods Notes Exercises

Generalized Analytic Functions

Preliminaries

Algebras associated with groups

A theorem of Bochner

Generalized analytic functions

Analytic measures

Local product decomposition

The Hardy spaces

Weak-star maximality

Weight functions

Invariant subspaces

Structure of cocycles

Cocycles and invariant subspaces Notes Exercises

Analytic Capacity and Rational Approximation

Analytic capacity

Elements of analytic capacity

Continuous analytic capacity

Peaking criteria

Criteria for $R(K)=C(K)$

Analytic diameter

A scheme for approximation

Criteria for $R(K)=A(K)$

Failure of approximation

Pointwise bounded approximation

Pointwise bounded approximation with same norm

Estimates for integrals

Analytically negligible sets Notes Exercises

Bibliography

List of special symbols

Index

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