Beschreibung
Good+. Cloth hardcover, xii + 182 pages, NOT ex-library. Good+ copy, signs of minor wear. Book is mildly age-toned, clean, with unmarked text, free of inscriptions and stamps, firmly bound. Boards show faint fingermarks. No dust jacket. -- Banach algebras are Banach spaces equipped with a continuous binary operation of multiplication. Numerous spaces considered in functional analysis are also algebras, e.g. the space C (0, 1) with pointwise multiplication of functions, or the space l1 with convolution multiplication of sequences. Theorems of the general theory of Banach algebras, applied to those spaces, yield several classical results of analysis, e.g. the Wiener theorem and the Wiener-Levy theorem on trigonometric series, or theorems on the spectral theory of operators. The foundations of the theory of Banach algebras are due to Gelfand. It was his astonishingly simple proof of the Wiener theorem that first turned the attention of mathematicians to the new theory. Certain specific algebras had been studied before, e.g. algebras of endomorphisms of Banach spaces, or weak-closed subalgebras of the algebra of endomorphisms of Hilbert spaces (the so-called von Neumann algebras or Wx algebras); also certain particular results had been obtained earlier. But the first theorem of the general theory of Banach algebras was the theorem on the three possible forms of normal fields, announced by Mazur in 1938. This result, now known as the Gelfand-Mazur theorem, is the starting point of Gelfand's entire theory of Banach algebras. Mazur's original proof is included in this volume; it is its first publication. The reader of this book is supposed to have some knowledge of functional analysis, algebra, topology, analytic functions and measure theory. The book is intelligible to first year university students, though certain sections are based on material which grew beyond the usual program of university mathematics, e.g. Chapter III makes use of the theory of analytic functions of several variables. However, the necessary notions and facts are always given explicitly and provided with references. -- Contents: 1 General Information about Banach Algebras [Notation, Basic Definitions, Theorems; Topological Algebras and Banach Algebras; Examples of Banach Algebras; Methods of Construction ol Banach Algebras; Mazur-Gelfand Theorem; Some Remarks on Algebras without Unit; Group of Invertible Elements of a Banach Algebra]; 2 Commutative Banach Algebras [Multiplicative Linear Functionals; General Form of Multiplicative Linear Functionals in Specific Banach Algebras; Maximal Ideal Space; Structure Space; Spectral Norm and the Radical; Semi-simple Algebras; Topological Divisors of Zero; Shilov Boundary]; 3 Analytic Functions in Banach Algebras [Analytic Functions of One Complex Variable; Spectrum and the Joint Spectrum; Characterization of Maximal Ideals in Banach Algebras; Analytic Functions of Several Complex Variables; Decomposition of a Commutative Banach Algebra into a Direct Sum of Ideals; Locally Analytic Operations. Eva Kallin's Counter-Example; Regular Algebras]; 4 Algebras with an Involution [Introductory Remarks; B*-algebras & Gelfand-Naimark Theorem; Theory of Representations of Algebras with an Involution; Representation Theorem for B*-algebras; Some Consequences of the Representation Theorem; Irreducible Representations and Non-decomposable Positive Functionals]; 5 Function Algebras [Realizations; Isometries; Bishop Boundary & Choquet Boundary; Dirichlet Algebras; Antisymmetric Algebras]; Appendix; References; Index of symbols; Author index; Subject index.
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