Banach Algebras and Automatic Continuity

by Dales, H. Garth

• ISBN13: 9780198500131
• ISBN10: 0198500130
• Format: Hardcover
• Copyright: 2001-05-17
• Publisher: Oxford University Press

Summary

This volume expounds the general theory of Banach algebras and shows how their topology is often determined by their algebraic structures. It synthesizes work in this area from the past 20 years, including many new and unpublished results, and probes the questions of when homomorphisms and derivations from Banach algebras are automatically continuous and seeks canonical forms for these maps. Many specific classes of Banach algebras are described, including function algebras,group algebras, algebras of perators, C*-algebras, and radical Banach algebras. This is essential reading for anyone interested in Banach algebras as it interweaves algebra, functional analysis, complex analysis, set theory and logic.

Table of Contents

Algebraic foundations	1	(150)
Foundations and ordered sets	2	(9)
Semigroups	11	(13)
Algebras, ideals, and fields	24	(26)
Modules and primitive ideals	50	(18)
Radicals and spectra	68	(13)
Polynomials and formal power series	81	(14)
Valuation algebras	95	(19)
Derivations	114	(13)
Cohomology	127	(15)
Involutions	142	(9)
Banach and topological algebras	151	(190)
Introduction to normed algebras	151	(16)
Topological algebras	167	(24)
Spectra and Gel'fand theory	191	(19)
The functional calculus	210	(19)
Banach algebras of operators	229	(9)
Banach modules	238	(23)
Intertwining maps and derivations	261	(12)
Cohomology	273	(33)
Bounded approximate identities and factorization	306	(35)
Banach algebras with an involution	341	(65)
General theory	341	(9)
C*-algebras	350	(21)
Group algebras	371	(35)
Commutative Banach algebras	406	(190)
Algebras of functions	406	(26)
Algebras of continuous functions	432	(15)
Uniform algebras	447	(11)
Algebras of differentiable and Lipschitz functions	458	(22)
Abelian group algebras	480	(19)
Banach algebras of power series	499	(19)
Convolution algebras on the real line	518	(40)
Prime ideals	558	(11)
Classification of commutative, radical Banach algebras	569	(12)
Commutative topological algebras	581	(15)
Automatic continuity theory	596	(189)
Uniqueness of norm	597	(12)
The separating space and the stability lemma	609	(26)
The continuity ideal and the singularity set	635	(25)
The main boundedness theorem	660	(34)
Linear functionals	694	(14)
Continuous and discontinuous derivations	708	(54)
Embedding algebras in Banach algebras	762	(23)
Appendix	785	(95)
A.1 Topological and metric spaces	785	(9)
A.2 Complex analysis	794	(16)
A.3 Functional analysis	810	(24)
A.4 Measures and integrals	834	(7)
A.5 Analytic spaces	841	(39)
Index of symbols	880	(6)
Index of theorems, lemmas, identities, and examples	886	(4)
Index	890