Function Algebras

1 Preliminaries.- 1.1. Commutative Banach algebras.- 1.2. Measures.- 1.3. Convexity.- 1.4. Holomorphic functions of several complex variables.- Notes.- 2 Boundaries.- 2.1. Function algebras.- 2.2. Representing measures.- 2.3. The Choquet boundary.- 2.4. The Shilov boundary.- 2.5. Geometric characterization.- 2.6. Representing theorems.- Notes.- 3 Algebras on the maximal ideal space.- 3.1. The maximal ideal space.- 3.2. Locally analytic functions.- 3.3. The local maximum modulus principle.- 3.4. Gleason parts.- Notes.- 4 Approximation and interpolation.- 4.1. Restrictions.- 4.2. The case of the intersections of peak sets.- 4.3. Antisymmetry.- 4.4. Some characterizations of C(X).- Notes.- 5 Hp-spaces.- 5.1. Definitions and basic lemmas.- 5.2. The theorem of F. and M. Riesz and Szego theorem.- 5.3. The factorization theorem.- 5.4. The characterization of the functions in Hp.- 5.5. Invariant subspaces.- 5.6. The algebra H?(dm).- Notes.- 6 Special classes of function algebras.- 6.1. Dirichlet and logmodular algebras.- 6.2. Algebras generated by inner functions.- 6.3. Maximal algebras.- 6.4. Functions algebras on compact sets of the complex plane.- 6.5. The standard algebra and H? algebra.- Notes.- 7 Operator representations of function algebras.- 7.1. Positive definite maps on C(X). Spectral and semispectral measures.- 7.2. Representations of function algebras ..- 7.3. Representations of the algebra C(X).- Notes.- 8 Elements of spectral theory of representations of function algebras.- 8.1. The canonical decomposition.- 8.2. The spectral dilation and attached spectral measures.- 8.3. Szego measures and natural representations.- 8.4. The Wold decomposition.- 8.5. Decompositions with respect to Gleason parts.- Notes.- 9 Elements of prediction theory on S- generated algebras.- 9.1. Semigroups of contractions.- 9.2. The Wold decomposition.- 9.3. The semigroup of unilateral translations.- 9.4. Representations of S-generated algebras.- 9.5. Prediction theorems.- Notes.- 10 Some examples in the spectral theory of non-normal operators.- 10.1. The case of a single contraction.- 10.2. Operators having spectral sets with connected complement.- 10.3. Finite systems of commuting contractions.- Notes.- Reference list.