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Abstract
In this Ph.D. thesis we study the relationship between the algebraic structure and the Banach space properties of uniform algebras. Banach algebras and in particular uniform algebras has been studied extensively and continue to be an active research area. There are many reasons for this strong and continuous interest. One can notice that uniform algebras has been a natural setting for areas like Classical Analysis, Functional Analysis and Approximation Theory.
Being a Banach space every Banach algebra is naturally related to its dual space. Linear operators have been used in many instances to characterize different Banach spaces, and Banach algebras are no exception. Among many different ways in using linear operators for such characterizations, there is one particularly useful, namely the method of factoring of operators. The idea behind this method is that when one can properly factor an operator through a well known Banach space, the Banach space that is the domain for the operator inherits some of the properties of the space used in the factorization.
Discrete linear operators form an important class of operators. One can point out many different reasons why this is so, but we want to emphasize that in almost every practical application people deal with discrete linear operators. Let us just mention the problems related to polynomial and spline interpolation. In general, the methods for recovering a function from a given discrete set of its values have attracted constant attention. One very important question that needs to be addressed when applying such methods is the question of their limitations. The dissertation addresses this question in a quite general setting. It is pointed out that the answer one gets may depend strongly on the interplay between the Banach space properties and the algebraic structure of a given uniform algebra. New results are obtained and their generality and importance are illustrated by several examples.
Chapter 1 contains an introduction to the theory of Banach algebras with an emphasis on those parts of the theory that deal with uniform algebras. Also we discuss some particular aspects of the general Banach Space Theory. The classical example of the disk algebra is considered as well. The importance of this example is explained by presenting two well known theorems that serve as models when considering the general case of uniform algebras.
In Chapter 2 we consider the question of using discrete linear operators as a tool for recovery of the elements of the disk algebra. We introduce a special class of sets in the unit disk and state a necessary condition for a solution to that recovery problem. Using this class of sets we narrow the gap between the existing sufficient and necessary conditions, and in some particular cases we are able to give a complete characterization for the recovery problem.
In Chapter 3 we investigate the recovery question in its full generality. We emphasize the importance of the concept of the boundary of a uniform algebra, and prove a general result. We provide various applications our main result. Some of them reprove in a simpler way well known theorems others appear to be new.