Topological Algebras
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Table of contents
- Cover
- Contentsvii
- Prefaceix
- Chapter 1. Algebraic Preliminaries1
- § 1. Some Basic Concepts and Results1
- § 2. Ideals and Radical12
- § 3. Characters and Hypermaximal Ideals22
- § 4. Extensions of Ideals30
- § 5. Regular Representation and Primitive Ideal36
- § 6. Real and Complex Algebras43
- § 7. Spectrum and Quasi-spectrum52
- § 8. Extended Spectrum and Extended Quasi-spectrum61
- § 9. Strictly Real Algebras67
- Chapter 2. Topological Preliminaries73
- § 1. Topological Groups and Linear Spaces73
- § 2. Topological Algebras84
- § 3. Completions of Topological Linear Spaces and Topological Algebras91
- Chapter 3. Some Types of Topological Algebras100
- § 1. Quarter-norms100
- § 2. ρ-Seminorms110
- § 3. Quarternormed Algebras.(F) Algebras116
- § 4. ρ-Seminormed Algebras; ρ-Banach Algebras131
- § 5. Bounded Linear Transformations on ρ -Seminormed Linear Spaces140
- § 6. Topological Algebras with Inverses148
- § 7. Topological Zero Divisors158
- Chapter 4. Locally Pseudo-Convex Spaces and Algebras174
- § 1. ρ -Convexity174
- § 2. Locally Bounded Algebras185
- § 3. Locally Pseudo-Convex Spaces189
- § 4. Locally Pseudo-Convex Algebras195
- § 5. Projective Limit Decomposition201
- § 6. Metrizable Locally Pseudo-Convex Algebras207
- § 7. Ample Algebras213
- § 8. Topological Spectral Radius216
- Chapter 5. Some Analysis222
- § 1. Vector-valued Differentiability and Analyticity222
- § 2. Exponential and Logarithmic Vector Functions227
- § 3. Square Roots and Quasi-square Roots234
- § 4. Complex Vector-valued Line Integrals and Cauchy's Theorems239
- § 5. Power Series Operations in Topological Algebras253
- Chapter 6. Spectral Analysis in Topological Algebras262
- § 1. Spectral Properties262
- § 2. The Resolvent Function264
- § 3. The Pseudo-Resolvent Function275
- § 4. Spectral Algebras282
- § 5. Gelfand-Mazur and Other Similar Theorems284
- § 6. Turpin's Theorem on Locally Convex Algebras290
- Chapter 7. Gelfand Representation Theory296
- § 1. Ideals of Topological Algebras296
- § 2. Gelfand Algebras303
- § 3. The Gelfand Representation311
- § 4. GB Algebras324
- § 5. Holomorphic Functional Calculus for a Single Algebra Element335
- § 6. Automorphisms and Derivations344
- Chapter 8. Commutative Topological Algebras353
- § 1. Function Algebras353
- § 2. Shilov Boundary361
- § 3. Hull-Kernel Topology370
- § 4. Completely Regular Algebras377
- § 5. Holomorphic Functional Calculus for Several Commutative Algebra Elements388
- § 6. Shilov Idempotent Theorem403
- Chapter 9. Norm Uniqueness Theorems411
- § 1. Norm-uniqueness Theorem of Gelfand411
- § 2. Rickart Seperating Function412
- § 3. Topological Modules416
- § 4. Norm-Uniqueness Theorems for Non-commutative Algebras419
- Appendix427
- Type Chart429
- Bibliography431
- Index435
- List of Special Symbols443
- List of Special Abbreviations445
This book consists of nine chapters. Chapter 1 is devoted to algebraic preliminaries. Chapter 2 deals with some of the basic definition and results concerning topological groups, topological linear spaces and topological algebras. Chapter 3 considered some generalizations of the norm. Chapter 4 is concerned with a generalization of the notion of convexity called p-convexity. In Chapter 5 some differential and integral analysis involving vector valued functions is developed. Chapter 6 is concerned with spectral analysis and applications. The Gelfand representation theory is the subject-matter of Chapter 7. Chapter 8 deals with commutative topological algebras. Finally in Chapter 9 an exposition of the norm uniqueness theorems of Gelfand and Johnson (extended to p-Banach algebras) is given.