Banach Algebras and the General Theory of *-Algebras: Volume 1, Algebras and Banach Algebras (Encyclopedia of Mathematics and its Applications, Series Number 49)

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 emphasizes the role of *-algebraic structure and explores the algebraic results that underlie the theory of Banach algebras and *-algebras. The first volume, which contains previously unpublished results, is an independent, self-contained reference on Banach algebra theory.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 of complex algebras rather than topological algebras. Proofs are presented in complete detail at a level accessible to graduate students. The book contains a wealth of historical comments, background material, examples, particularly in noncommutative harmonic analysis, and an extensive bibliography. Volume II is forthcoming. Front matter ......Page 1 Preface ......Page 3 Contents ......Page 7 1 Introduction to Normed Algebras; Examples ......Page 11 1.1 Norms and Semi-norms on Algebra s ......Page 12 1.2 Double Centralizers and Extensions ......Page 34 1.3 Sums, Products and Limits ......Page 47 1.4 Arens Multiplication ......Page 56 1.5 Algebras of Functions ......Page 74 1.6 Matrix Algebras ......Page 79 1.7 Operator Algebras ......Page 88 1.8 Group Algebras on T ......Page 121 1.9 Group Algebras ......Page 138 1.10 Tensor Products ......Page 170 2 The Spectrum ......Page 197 2.1 Definition of the Spectrum ......Page 202 2.2 Spectral Semi-norms ......Page 218 2.3 The Jacobson Radical and the Fundamental Theorem ......Page 234 2.4 Spectral Algebras ......Page 243 2.5 Spectral Subalgebras and Topological Divisors of Zero ......Page 255 2.6 Numerical Range in Banach Algebras ......Page 270 2.7 The Spectrum in Finite-Dimensional Algebras ......Page 282 2.8 Spectral Theory of Operators ......Page 296 2.9 Topological Algebras ......Page 307 3 Commutative Algebras and Functional Calculus ......Page 311 3.1 Gelfand Theory ......Page 313 3.2 Silov Boundary, Hulls and Kernels ......Page 329 3.3 Functional Calculus ......Page 351 3.4 Examples and Applications of Functional Calculus ......Page 366 3.5 Multivariable Functional Calculus ......Page 388 3.6 Commutative Group Algebras ......Page 427 4 Ideals, Representations and Radicals ......Page 447 4.1 Ideals and Representations ......Page 450 4.2 Representations and Norms ......Page 466 4.3 The Jacobson Radical ......Page 484 4.4 The Baer Radical ......Page 491 4.5 The Brown-McCoy or Strong Radical ......Page 500 4.6 Subdirect Products ......Page 505 4 7 Categorical Theory of Radicals ......Page 508 4.8 History of Radicals and Examples ......Page 513 5.1 Approximate Identities and Examples ......Page 529 5.2 General Factorization Theorems ......Page 544 5.3 Countable Factorization Theorems ......Page 553 6 Automatic Continuity ......Page 563 6.1 Automatic Continuity of Homornorphisms into A ......Page 565 6.2 Automatic Continuity of Homornorphisms from A ......Page 581 6.3 Jordan Homornorphisms ......Page 589 6.4 Derivations ......Page 601 6.5 Jordan Derivations ......Page 624 7.1 The Huii-Keniel Topology ......Page 629 7.2 Completely Regular Algebras ......Page 642 7.3 Primary Ideals and Spectral Synthesis ......Page 649 7.4 Strongly Harmonic Algebras ......Page 655 8 Algebras with Minimal Ideals ......Page 665 8.1 Finite-Dimensional Algebras ......Page 667 8.2 Minimal Ideals and the Socle ......Page 677 8.3 Algebras of Operators with Minimal Ideals ......Page 682 8.4 Modular Annihilator Algebras ......Page 688 8.5 Fredholrn Theory ......Page 698 8.6 Algebras with Countable Spectrum for Elements ......Page 705 8.7 Classes of Algebras with Large Socle ......Page 709 8.8 Examples ......Page 721 Bibliography ......Page 725 Index ......Page 793 Symbol Index ......Page 801 V. 1. Algebras And Banach Algebras -- V. 2. Algebras. Theodore W. Palmer. Includes Bibliographical References. v. 1. Algebras and Banach algebras 2. [without special title].