Abstract
The book, “Big Planes, Boundaries, and Function Algebras,” discusses the latest developments made in analytic almost periodic function theory in a contemporary uniform algebra setting. Analytic Γ-almost-periodic functions in domains of the complex plane are presented as Γ -analytic functions in corresponding domains of the big-plane. Definitions, various descriptions, and general properties of uniform algebras of Γ -analytic functions on big-plane subsets and especially of the big-disc algebra, and algebras of analytic Γ -almost-periodic functions are presented in the chapter. It also discusses the properties of Γ -analytic functions defined on arbitrary subsets in the big-plane and in particular in the big-disc. The important notions of spectral mapping and spectrum of a multiplicative subsemigroup of a commutative Banach algebra are introduced in the chapter. It also discusses various characterizations and basic properties of multi-tuple Shilov boundaries of function spaces.
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