Shift invariant uniform algebras on groups

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Shift-invariant algebras are uniform algebras of continuous functions defined on compact connected groups, invariant under shifts by group elements. They stem from generalized analytic functions introduced by Arens and Singer nearly fifty years ago and are the focus of this work. The text also explores associated algebras of almost periodic functions and bounded analytic functions on the unit disc within the shift-invariant framework. This approach yields fresh perspectives, novel questions, and surprising properties. The book primarily presents recent and some unpublished results, with foundational concepts originating from earlier work. Chapter 1 introduces basic terminology and standard properties of uniform algebras, discussing associated algebras like Bourgain algebras and polynomial extensions, concluding with conditions for mappings between uniform algebras to be algebraic isomorphisms. Chapter 2 covers fundamentals and properties of three classical function families: almost periodic functions, harmonic functions, and H-functions on the unit circle, with extensions to arbitrary compact groups in Chapter 7. Chapter 3 surveys basic properties of topological groups, their characters, dual groups, and introduces the crucial concepts of weak and strong hulls of a semigroup.