Function Algebras

ISBN-13: 9789028604452 / Angielski / Miękka / 1975 / 272 str.

I. Suciu

Opis

Under the title of Function Algebras we may now include a very large number of works. published mainly in the last decade, which consti- tute one of the important chapters of functional analysis. This chapter has grown up from various problems. permanently furnished to mathe- matics. by the theory of functions. using modern methods of algebra, topology and functional analysis and presenting large possibilities of applications in operators theory. Herefrom proceeds its living character, the variety of obtained results. the variety of forms and contexts in which these results can be found. This also explains the difficulty of an exhaustive exposition of these problems. The purpose of the monograph is to present a coherent exposition of the fundamental results of this theory with an orientation to their applicability to the theory of operator representations of function alge- bras. The idea of such a work appeared during the seminaries on function algebras held at the Mathematical Institute in Bucharest. under the direc- tion of C. Foia~ and at the Faculty of Mathematics and Mechanics under the direction of N. Boboc. It is a pleasure for the author to express his gratitude to C. Foia~ for assistance in his efforts. in general. and for the large contribution the discussions and cooperation with him had brought in the elaboration of this monograph. I also would like to thank N. Boboc for the clear discussions we have had during the seminaries and the elaboration of some chapters.

Spis treści

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 Szegö 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. Szegö 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.