The central subject of this book - the theory of shift-invariant algebras - is an outgrowth of the established theory of generalized analytic functions. This general framework carries along associated subalgebras of almost periodic functions of real variables and of bounded analytic functions on the unit disc. The text describes Bourgain algebras, orthogonal measures, and primary ideals of big disc algebras. The notion of a harmonic function is extended on compact abelian groups, and corresponding Fatou-type theorems are proven. Finally, the book introduces and studies classes of inductive limits of standard uniform algebras, including Blasche algebras. This is the first monograph devoted entirely to the topic.

Klappentext

This book on the theory of shift-invariant algebras is the first monograph devoted entirely to an outgrowth of the established theory of generalized analytic functions on compact groups. Associated subalgebras of almost periodic functions of real variables and of bonded analytic functions on the unit disc are carried along within the general framework.

Zusammenfassung
Shift-invariant algebras are uniform algebras of continuous functions de?ned on compactconnectedgroups,thatareinvariantundershiftsbygroupelements. They areoutgrowths of generalized analytic functions, introduced almost ?fty yearsago by Arens and Singer, and are the central object of this book. Associated algebras of almost periodic functions of real variables and of bounded analytic functions on the unit disc are also considered and carried along within the shift-invariant framework. The adopted general approach leads to non-standard perspectives, never-asked-before questions, and unexpected properties. Thebookisbasedmainlyonourquiterecent,someevenunpublished,results. Most of its basic notions and ideas originate in [T2]. Their further development, however, can be found in journal or preprint form only. Basic terminologyand standard properties of uniform algebrasarepresented in Chapter 1. Associated algebras, such as Bourgain algebras, polynomial ext- sions, and inductive limit algebras are introduced and discussed. At the end of the chapter we present recently found conditions for a mapping between uniform algebras to be an algebraic isomorphism. In Chapter 2 we give fundamentals, v- ious descriptions and standard properties of three classical families of functions p almost periodic functions of real variables, harmonic functions, andH -functions on the unit circle. Later on, in Chapter 7, we return to some of these families and extend them to arbitrary compact groups. Chapter 3 is a survey of basic prop- ties of topological groups, their characters, dual groups, functions and measures on them. We introduce also the instrumental for the sequel notion of weak and strong hull of a semigroup.

Inhalt
Banach algebras and uniform algebras.- Three classical families of functions.- Groups and semigroups.- Shift-invariant algebras on compact groups.- Extension of semicharacters and additive weights.- G-disc algebras.- Harmonicity on groups and G-discs.- Shift-invariant algebras and inductive limit algebras on groups.