In this post, we will see the book *Commutative Normed Rings* by I. Gelfand; D. Raikov; G. Shilov.

# About the book

The present book gives an account of the theory of commutative normed rings with applications to analysis and topology. The paper by I. N. Gelfand and M. A. Naimark Normed Rings with an Involution and their Representations, which is presented here as Chapter VIII, may serve as an introduction to the theory of non-commutative normed rings with an involution.

The book is addressed to mathematicians—students in advanced courses, research students, and scholars—who are interested in functional analysis and its applications.

The book was translated from Russian and was published in 1964.

Credits to original uploader.

You can get the book here.

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# Contents

# PART ONE

## I. THE GENERAL THEORY OF COMMUTATIVE NORMED RINGS 15

§ 1. The Concept of a Normed Ring 15

§ 2. Maxiimal Ideals 20

§ 3. Abstract Analytic Functions 27

§ 4. Functions on Maximal Ideals. The Radical of a Ring 30

§ 5. The Space of Maximal Ideals 37

§ 6. Analytic Functions of an Element of a Ring 46

§ 7. The Ring R of Functions x(M) 51

§ 8. Rings with an Involution 56

## II. THE GENERAL THEORY OF COMMUTATIVE NORMED RINGS (cont’d)66

§ 9. The Connection between Algebraic and Topological Isomorphisms 66

§ 10. Generalized Divisors of Zero 69

§ 11. The Boundary of the Space of Maximal Ideals 73

§ 12. Extension of Maximal Ideals 78

§ 13. Locally Analytic Operations on Certain Elements of a Ring 80

§ 14. Decomposition of a Normed Ring into a Direct Sum of Ideals 94

§ 15. The Normed Space Adjoint toa Normed Ring 97

# PART TWO

## III. THE RING OF ABSOLUTELY INTEGRABLE FUNCTIONS AND

THEIR DISCRETE ANALOGUES 100

§ 16. The Ring V of Absolutely Integrable Functions on the Line 100

§ 17. Maximal Ideals of the Rings V and V+ 106

§ 18. The Ring of Absolutely Integrable Functions With a Weight 113

§ 19. Discrete Analogues to the Rings of Absolutely Integrable

Functions 116

## IV. HARMONIC ANALYSIS ON COMMUTATIVE LOCALLY COMPACT GROUPS 121

§ 20. The Group Ring of a Commutative Locally Compact Group 123

§ 21. Maximal Ideals of the Group Ring and the Characters of

a Group 129

§ 22. The Uniqueness Theorem for the Fourier Transform and the Abundance of the Set of Characters 135

$ 23. The Group of Characters 141

§ 24. The Invariant Integral on the Group of Characters 144

§ 25. Inversion Formulas for the Fourier Transform 151

§ 26. The Pontrjagin Duality Law 156

§ 27. Positive-Definite Functions 159

## V. THE RING OF FUNCTIONS OF BOUNDED VARIATION ON A LINE 165

§ 28. Functions of Bounded Variation on a Line 165

§ 29. The Ring of Jump Functions 167

§ 30. Absolutely Continuous and Discrete Maximal Ideals of the Ring) 176

§ 31. Singular Maximal Ideals of the Ring V^(b) 180

§ 32. Perfect Sets with Linearly Independent Points. The Asymmetry of the Ring V^(b) 187

§ 33. The General Form of Maximal Ideals of the Ring V^(b) 192

# PART THREE

## VI. REGULAR RINGS 197

§ 34. Definitions, Examples, and Simplest Properties 197

§ 35: The Local Theorem 200

§ 36. Minimal Ideals 204

§ 37. Primary Ideals 205

§ 38. Locally Isomorphic Rings 207

§ 39. Connection between the Residue-Class Rings of Two Rings of Functions, One Embedded in the Other 210

§ 40. Wiener’s Tauberian Theorem 213

§ 41. Primary Ideals in Homogeneous Rings of Functions 214

§ 42. Remarks on Arbitrary Closed Ideals. An Example of L. Schwartz 219

## VII. RINGS WITH UNIFORM CONVERGENCE 223

§ 43. Symmetric Subrings of C(S) and Compact Extensions of Space S 223

§ 44. The Problem of Arbitrary Closed Subrings of the Ring C(S) 227

§ 45. Ideals in Rings with Uniform Convergence 234

## VII. NORMED RINGS WITH AN INVOLUTION AND THEIR REPRESENTATIONS 240

§ 46. Rings with an Involution and their Representations 241

§ 47. Positive Functionals and their Connection with Representations Of Rings 244

§ 48. Embedding of a Ring with an Involution in a Ring of Operators 251

§ 49. Indecomposable Functionals and Irreducible Representations 255

§ 50. The Case of Commutative Rings 259

§ 51. Group Rings 263

§ 52. Example of an Unsymmetric Group Ring 268

## IX. THE DECOMPOSITION OF A COMMUTATIVE NORMED RING INTO A DIRECT SUM OF IDEALS 275

§ 53. Introduction 275

§ 54. Characterization of the Space of Maximal Ideals of a Commutative Normed Ring 277

§ 55. A Problem on Analytic Functions in a Finitely Generated Ring 278

§ 56. Construction of a Special Finitely Generated Subring 282

§ 57. Proof of the Theorem on the Decomposition of a Ring

into a Direct Sum of Ideals 285

S56. Some Corollaries 285

HISTORICO-BIBLIOGRAPHICAL NOTES 291

BIBLIOGRAPHY 295

INDEX 303