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This is the first volume of a two volume set that provides a modern account of basic Banach algebra theory including all known results on general Banach *-algebras. This account emphasises the role of *-algebra structure and explores the algebraic results which underlie the theory of Banach algebras and *-algebras. This first volume is an independent, self-contained reference on Banach algebra theory. Each topic is treated in the maximum interesting generality within the framework of some class complex algebras rather than topological algebras. In both volumes proofs are presented in complete detail at a level accessible to graduate students. In addition, the books contain a wealth of historical comments, background material, examples, particularly in noncommutative harmonic analysis, and an extensive bibliography. Together these books will become the standard reference for the general theory of *-algebras.

Operator theory --- Banach algebras --- Banach algebras. --- Algebras, Banach --- Banach rings --- Metric rings --- Normed rings --- Banach spaces --- Topological algebras --- Algebres de banach --- Algebres topologiques --- C star algebres

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Functional analysis --- Banach algebras --- 517.986 --- Algebras, Banach --- Banach rings --- Metric rings --- Normed rings --- Banach spaces --- Topological algebras --- Topological algebras. Theory of infinite-dimensional representations --- Banach algebras. --- 517.986 Topological algebras. Theory of infinite-dimensional representations --- Banach, Algèbres de

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Functional analysis --- Banach algebras. --- Banach, Algèbres de --- Ideals (Algebra) --- Idéaux (algèbre) --- Modules (Algebra) --- Modules (algèbre) --- Continuity. --- Banach algebras --- Continuity --- Finite number systems --- Modular systems (Algebra) --- Algebra --- Finite groups --- Rings (Algebra) --- Algebraic ideals --- Algebraic fields --- Continuum --- Mathematics --- Indivisibles (Philosophy) --- Algebras, Banach --- Banach rings --- Metric rings --- Normed rings --- Banach spaces --- Topological algebras --- Philosophy --- Banach, Algèbres de.

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Harmonic analysis. Fourier analysis --- Banach algebras --- Hardy classes --- Invariant subspaces --- Riemann surfaces --- Surfaces, Riemann --- Functions --- Subspaces, Invariant --- Functional analysis --- Hilbert space --- Classes, Hardy --- Hp classes --- Function algebras --- Functions of complex variables --- Algebras, Banach --- Banach rings --- Metric rings --- Normed rings --- Banach spaces --- Topological algebras --- Riemann surfaces. --- Hardy classes. --- Invariant subspaces. --- Banach algebras. --- Banach, Algèbres de. --- Sous-espaces invariants. --- Hardy, Classes de. --- Riemann, Surfaces de.

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Analytical spaces --- Function algebras --- Banach algebras --- Algèbres de fonctions --- Banach, Algèbres de --- 517.98 --- Algebras, Function --- Analytic functions --- Algebras, Banach --- Banach rings --- Metric rings --- Normed rings --- Banach spaces --- Topological algebras --- Functional analysis and operator theory --- Banach algebras. --- Function algebras. --- 517.98 Functional analysis and operator theory --- Algèbres de fonctions --- Banach, Algèbres de --- Fonctions de plusieurs variables complexes