Uniform Frechet Algebras (eBook)

H. Goldmann (Autor)

eBook Download: PDF

1990 | 1. Auflage
354 Seiten
Elsevier Science (Verlag)
978-0-08-087273-5 (ISBN)

Infos

The first part of this monograph is an elementary introduction to the theory of Fré,chet algebras. Important examples of Fré,chet algebras, which are among those considered, are the algebra of all holomorphic functions on a (hemicompact) reduced complex space, and the algebra of all continuous functions on a suitable topological space.

The problem of finding analytic structure in the spectrum of a Fré,chet algebra is the subject of the second part of the book. In particular, the author pays attention to function algebraic characterizations of certain Stein algebras (= algebras of holomorphic functions on Stein spaces) within the class of Fré,chet algebras.

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The first part of this monograph is an elementary introduction to the theory of Frechet algebras. Important examples of Frechet algebras, which are among those considered, are the algebra of all holomorphic functions on a (hemicompact) reduced complex space, and the algebra of all continuous functions on a suitable topological space.The problem of finding analytic structure in the spectrum of a Frechet algebra is the subject of the second part of the book. In particular, the author pays attention to function algebraic characterizations of certain Stein algebras (= algebras of holomorphic functions on Stein spaces) within the class of Frechet algebras.

Inhalt

Front Cover 1
Uniform Frechet Algebras 4
Copyright Page 5
CONTENTS 12
Preface 8
PART 1. BANACH ALGEBRAS, ALGEBRAS OF HOLOMORPHIC FUNCTIONS, AN INTRODUCTION 16
CHAPTER 1. An excurs on Banach algebras 18
(1.1) General theory 18
(1.2) The spectrum of a B-algebra 24
(1.3) The Shilov boundary 33
(1.4) The holomorphic functional calculus 40
(1.5) Analytic structure in spectra 45
CHAPTER 2. The algebra of holomorphic functions 48
(2.1) General theory 49
(2.2) Analytic continuation 56
(2.3) Stein spaces 65
PART 2. GENERAL THEORY OF FRECHET ALGEBRAS 72
CHAPTER 3. Theory of Fréchet algebras, basic results 74
(3.1) Frechet algebras 74
(3.2) The spectrum of a Fréchet algebra 86
(3.3) Projective limits 99
CHAPTER 4. General theory of uniform Fréchet algebras 106
(4.1) Uniform Fréchet algebras 106
(4.2) Extension of a uniform Fréchet algebra 112
(4.3) Convexity for uniform Fréchet algebras 118
(4.4) Uniform Fréchet algebras with locally compact spectrum 124
CHAPTER 5. Finitely generated Fréchet algebras 128
(5.1) Finitely generated Fréchet algebras 128
(5.2) Rationally finitely generated Fréchet algebras 137
CHAPTER 6. Applications of the projective limit representation 144
(6.1) The holomorphic functional calculus 144
(6.2) The theorem of Arens-Royden 156
CHAPTER 7. AN Fréchet algebra whose spectrum is not a k-space 162
(7.1) The example of Dors 162
(7.2) Surjectivity of the transpose map 165
(7.3) The weak Nullstellensatz 170
CHAPTER 8. Semisimple Fréchet algebras 176
(8.1) Uniqueness of topology 176
(8.2) Continuity of derivations 180
CHAPTER 9. Shilov boundary and peak points for Fréchet algebras 186
(9.1) The Shilov boundary of a Fr6chet algebra 186
(9.2) Peak points for Fréxhet algebras 195
CHAPTER 10. Michael's problem 200
(10.1) Results on the automatic continuity of characters 200
(10.2) The approach of Dixon and Esterle 205
PART 3. ANALYTIC STRUCTURE IN SPECTRA 210
CHAPTER 11. Stein algebras 212
(11.1) Analytic structure in spectra of uF-algebras 212
CHAPTER 12. Characterizing some particular Stein algebras 220
(12.1) Polynomially convex analytic subsets 220
(12.2) Polynomially convex open subsets of Cn 223
(12.3) Open subsets of the plane 228
(12.4) Domains of holomorphy 230
(12.5) Logarithmically convex complete Reinhardt domains 234
CHAPTER 13. Liouville algebras 238
(13.1) Liouville algebras 238
CHAPTER 14. Maximum modulus principle 246
(14.1) Maximum modulus algebras 246
(14.2) Maximum modulus algebras and subharmonicity 252
CHAPTER 15. Maximum modulus algebras and analytic structure 258
(15.1) Riemann domains over Cn 258
(15.2) Maximum modulus algebras and finite mappings 263
CHAPTER 16. Higher Shilov boundaries 278
(16.1) Higher Shilov boundaries for uB-algebras 278
(16.2) Higher Shilov boundaries for uF-algebras 283
CHAPTER 17. Local analytic structure in the spectrum of a uniform Fréchet algebra 288
(17.1) Local rings of functions of uniform Fréchet algebras 288
CHAPTER 18. Reflexive uniform Fréchet algebras 300
(18.1) Reflexive uniform Fréchet algebras 300
(18.2) Gleason parts 302
CHAPTER 19. Uniform Fréchet Schwartz algebras 318
(19.1) Fréchet Schwartz algebras 319
(19.2) Strongly uniform Fréchet algebras 321
(19.3) Chevalley dimension for uniform Frvchet algebras 325
(19.4) A characterization of pure dimensional Stein algebras 330
(19.5) Riemann algebras 340
APPENDIX A. Subharmonic functions, Poisson integral 346
APPENDIX B. Functional analysis 350
List of symbols 351
References 353
Index 365

Produkt-Details

• Erscheint lt. Verlag: 17.4.1990
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik
• ISBN-10: 0-08-087273-5 / 0080872735
• ISBN-13: 978-0-08-087273-5 / 9780080872735