Uniform algebras

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 algebras 8. Compactifications 9. The algebra 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 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 5. Invertible elements of $H$ 6. Invariant subspaces 7. Embedding of analytic discs 8. Szego's theorem 9. Extremal functions in 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 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.