Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
Commutative Normed Rings
 
Commutative Normed Rings
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Softcover ISBN:  978-0-8218-2972-1
Product Code:  CHEL/170.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
Not yet published - Preorder Now!
Expected availability date: July 01, 2024
eBook ISBN:  978-1-4704-7789-9
Product Code:  CHEL/170.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $58.50
Softcover ISBN:  978-0-8218-2972-1
eBook: ISBN:  978-1-4704-7789-9
Product Code:  CHEL/170.S.B
List Price: $134.00 $101.50
AMS Member Price: $120.60 $91.35
Not yet published - Preorder Now!
Expected availability date: July 01, 2024
Commutative Normed Rings
Click above image for expanded view
Commutative Normed Rings
AMS Chelsea Publishing: An Imprint of the American Mathematical Society
Softcover ISBN:  978-0-8218-2972-1
Product Code:  CHEL/170.S
List Price: $69.00
MAA Member Price: $62.10
AMS Member Price: $62.10
Not yet published - Preorder Now!
Expected availability date: July 01, 2024
eBook ISBN:  978-1-4704-7789-9
Product Code:  CHEL/170.E
List Price: $65.00
MAA Member Price: $58.50
AMS Member Price: $58.50
Softcover ISBN:  978-0-8218-2972-1
eBook ISBN:  978-1-4704-7789-9
Product Code:  CHEL/170.S.B
List Price: $134.00 $101.50
AMS Member Price: $120.60 $91.35
Not yet published - Preorder Now!
Expected availability date: July 01, 2024
  • Book Details
     
     
    AMS Chelsea Publishing
    Volume: 1701964
    MSC: Primary 46

    From the Preface (1960):

    “This book is devoted to an account of one of the branches of functional analysis, the theory of commutative normed rings, and the principal applications of that theory. It is based on [the authors'] paper written ... in 1940, hard on the heels of the initial period of the development of this theory ...

    The book consists of three parts. Part one, concerned with the theory of commutative normed rings and divided into two chapters; the first containing foundations of the theory and the second dealing with more special problems. Part two deals with applications to harmonic analysis and is divided into three chapters. The first chapter discusses the ring of absolutely integrable functions on a line with convolution as multiplication and finds the maximal ideals of this ring and some of its analogues. In the next chapter, these results are carried over to arbitrary commutative locally compact groups and they are made the foundation of the construction of harmonic analysis and the theory of characters. A new feature here is the construction of an invariant measure on the group of characters and a proof of the inversion formula for Fourier transforms that is not based on theorems on the representation of positive-definite functions or positive functionals ... The last chapter of the second part—the most specialized of all the chapters—is devoted to the investigation of the ring of functions of bounded variation on a line with multiplication defined as convolution, including the complete description of the maximal ideals of this ring. The third part of the book is devoted to the discussion of two important classes of rings of functions: regular rings and rings with uniform convergence. The first of the chapters essentially studies the structure of ideals in regular rings. The chapter ends with an example of a ring of functions having closed ideals that cannot be represented as the intersections of maximal ideals. The second chapter discusses the ring \(C(S)\) of all bounded continuous complex functions on completely regular spaces \(S\) and various of its subrings ...

    Since noncommutative normed rings with an involution are important for group-theoretical applications, the paper by I. M. Gelfand and N. A. Naimark, ‘Normed Rings with an Involution and their Representations’, is reproduced at the end of the book, slightly abridged, in the form of an appendix ... This monograph also contains an account of the foundations of the theory of commutative normed rings without, however, touching upon the majority of its analytic applications ...

    The reader [should] have knowledge of the elements of the theory of normed spaces and of set-theoretical topology. For an understanding of the fourth chapter, [the reader should] also know what a topological group is. It stands to reason that the basic concepts of the theory of measure and of the Lebesgue integral are also assumed to be known ...”

    Readership

    Graduate students and research mathematicians.

  • Table of Contents
     
     
    • Front Cover
    • NOTE
    • PREFACE
    • TABLE OF CONTENTS
    • NOTE ON THE INTERDEPENDENCE OF THE CHAPTERS
    • PART ONE
    • CHAPTER I THE GENERAL THEORY OF COMMUTATIVE NORMED RINGS
    • § 1. The Concept of a Normed Ring
    • § 2. Maximal Ideals
    • § 3. Abstract Analytic Functions
    • § 4. Functions on Maximal Ideals. The Radical of a Ring
    • § 5. The Space of Maximal Ideals
    • § 6. Analytic Functions of an Element of a Ring
    • § 7. The Ring R of Functions x(M)
    • § 8. Rings With an Involution
    • CHAPTER II THE GENERAL THEORY OF COMMUTATIVE NORMED RINGS (continued)
    • § 9. The Connection between Algebraic and Topological Isomorphisms
    • § 10. Generalized Divisors of Zero
    • § 11. The Boundary of the Space of Maximal Ideals
    • § 12. Extension of Maximal Ideals
    • § 13. Locally Analytic Operations on Certain Elements of a Ring
    • § 14. Decomposition of a Normed Ring into a Direct Sum of Ideals
    • § 15. The Normed Space Adjoint to a Normed Ring
    • PART TWO
    • CHAPTER III THE RING OF ABSOLUTELY INTEGRABLE FUNCTIONS AND THEIR DISCRETE ANALOGUES
    • § 16. The Ring V of Absolutely Integrable Functions on the Line
    • § 17. Maximal Ideals of the Rings V and V +
    • § 18. The Ring of Absolutely Integrable Functions With a Weight
    • § 19. Discrete Analogues to the Rings of Absolutely Integrable Functions
    • CHAPTER IV HARMONIC ANALYSIS ON COMMUTATIVE LOCALLY COMPACT GROUPS
    • § 20. The Group Ring of a Commutative Locally Compact Group
    • § 21. Maximal Ideals of the Group Ring and the Characters of a Group
    • § 22. The Uniqueness Theorem for the Fourier Transform and the Abundance of the Set of Characters
    • § 23. The Group of Characters
    • § 24. The Invariant Integral on the Group of Characters
    • § 25. Inversion Formulas for the Fourier Transform
    • § 26. The Pontrjagin Duality Law
    • § 27. Positive-Definite Functions
    • CHAPTER V THE RING OF FUNCTIONS OF BOUNDED VARIATION ON A LINE
    • § 28. Functions of Bounded Variation on a Line
    • § 29. The Ring of Jump Functions
    • § 30. Absolutely Continuous and Discrete Maximal Ideals of the Ring V(b)
    • § 31. Singular Maximal Ideals of the Ring V(b)
    • § 32. Perfect Sets with Linearly Independent Points. The Asymmetry of the Ring V(b)
    • § 33. The General Form of Maximal Ideals of the Ring V(b)
    • PART III
    • CHAPTER VI REGULAR RINGS
    • § 34. Definitions, Examples, and Simplest Properties
    • § 35. The Local Theorem
    • § 36. Minimal Ideals
    • § 37. Primary Ideals
    • § 38. Locally Isomorphic Rings
    • § 39. Connection between the Residue-Class Rings of Two Rings of Functions, One Embedded in the Other
    • § 40. Wiener's Tauberian Theorem
    • § 41. Primary Ideals in Homogeneous Rings of Functions
    • § 42. Remarks on Arbitrary Closed Ideals. An Example of L. Schwartz
    • CHAPTER VII RINGS WITH UNIFORM CONVERGENCE
    • § 43. Symmetric Subrings of C ( S) and Compact Extensions of a Space S
    • § 44. The Problem of Arbitrary Closed Subrings of the Ring C(S)
    • § 45. Ideals in Rings with Uniform Convergence
    • CHAPTER VIII NORMED RINGS WITH AN INVOLUTION AND THEIR REPRESENTATIONS
    • § 46. Rings with an Involution and their Representations
    • § 47. Positive Functionals and their Connection with Representations of Rings
    • § 48. Embedding of a Ring with an Involution in a Ring of Operators
    • § 49. Indecomposable Functionals and Irreducible Representations
    • § 50. The Case of Commutative Rings
    • § 51. Group Rings
    • § 52. Example of an Unsymmetric Group Ring
    • CHAPTER IX THE DECOMPOSITlON OF A COMMUTATIVE NORMED RINGINTO A DIRECT SUM OF IDEALS
    • § 53. Introduction
    • § 54. Characterization of the Space of Maximal Ideals of a Commutative Normed Ring
    • § 55. A Problem on Analytic Functions in a Finitely Generated Ring
    • § 56. Construction of a Special Finitely Generated Subring
    • § 57. Proof of the Theorem on the Decomposition of a Ring into a Direct Sum of Ideals
    • § 58. Some Corollaries
    • HISTORICO-BIBLIOGRAPHICAL NOTES
    • HISTORICO-BIBLIOGRAPHICAL NOTES
    • BIBLIOGRAPHY
    • BIBLIOGRAPHY
    • BIBLIOGRAPHY FOR APPENDIX
    • BIBLIOGRAPHY TO SECOND APPENDIX
    • INDEX
    • INDEX
    • Back Cover
  • Additional Material
     
     
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1701964
MSC: Primary 46

From the Preface (1960):

“This book is devoted to an account of one of the branches of functional analysis, the theory of commutative normed rings, and the principal applications of that theory. It is based on [the authors'] paper written ... in 1940, hard on the heels of the initial period of the development of this theory ...

The book consists of three parts. Part one, concerned with the theory of commutative normed rings and divided into two chapters; the first containing foundations of the theory and the second dealing with more special problems. Part two deals with applications to harmonic analysis and is divided into three chapters. The first chapter discusses the ring of absolutely integrable functions on a line with convolution as multiplication and finds the maximal ideals of this ring and some of its analogues. In the next chapter, these results are carried over to arbitrary commutative locally compact groups and they are made the foundation of the construction of harmonic analysis and the theory of characters. A new feature here is the construction of an invariant measure on the group of characters and a proof of the inversion formula for Fourier transforms that is not based on theorems on the representation of positive-definite functions or positive functionals ... The last chapter of the second part—the most specialized of all the chapters—is devoted to the investigation of the ring of functions of bounded variation on a line with multiplication defined as convolution, including the complete description of the maximal ideals of this ring. The third part of the book is devoted to the discussion of two important classes of rings of functions: regular rings and rings with uniform convergence. The first of the chapters essentially studies the structure of ideals in regular rings. The chapter ends with an example of a ring of functions having closed ideals that cannot be represented as the intersections of maximal ideals. The second chapter discusses the ring \(C(S)\) of all bounded continuous complex functions on completely regular spaces \(S\) and various of its subrings ...

Since noncommutative normed rings with an involution are important for group-theoretical applications, the paper by I. M. Gelfand and N. A. Naimark, ‘Normed Rings with an Involution and their Representations’, is reproduced at the end of the book, slightly abridged, in the form of an appendix ... This monograph also contains an account of the foundations of the theory of commutative normed rings without, however, touching upon the majority of its analytic applications ...

The reader [should] have knowledge of the elements of the theory of normed spaces and of set-theoretical topology. For an understanding of the fourth chapter, [the reader should] also know what a topological group is. It stands to reason that the basic concepts of the theory of measure and of the Lebesgue integral are also assumed to be known ...”

Readership

Graduate students and research mathematicians.

  • Front Cover
  • NOTE
  • PREFACE
  • TABLE OF CONTENTS
  • NOTE ON THE INTERDEPENDENCE OF THE CHAPTERS
  • PART ONE
  • CHAPTER I THE GENERAL THEORY OF COMMUTATIVE NORMED RINGS
  • § 1. The Concept of a Normed Ring
  • § 2. Maximal Ideals
  • § 3. Abstract Analytic Functions
  • § 4. Functions on Maximal Ideals. The Radical of a Ring
  • § 5. The Space of Maximal Ideals
  • § 6. Analytic Functions of an Element of a Ring
  • § 7. The Ring R of Functions x(M)
  • § 8. Rings With an Involution
  • CHAPTER II THE GENERAL THEORY OF COMMUTATIVE NORMED RINGS (continued)
  • § 9. The Connection between Algebraic and Topological Isomorphisms
  • § 10. Generalized Divisors of Zero
  • § 11. The Boundary of the Space of Maximal Ideals
  • § 12. Extension of Maximal Ideals
  • § 13. Locally Analytic Operations on Certain Elements of a Ring
  • § 14. Decomposition of a Normed Ring into a Direct Sum of Ideals
  • § 15. The Normed Space Adjoint to a Normed Ring
  • PART TWO
  • CHAPTER III THE RING OF ABSOLUTELY INTEGRABLE FUNCTIONS AND THEIR DISCRETE ANALOGUES
  • § 16. The Ring V of Absolutely Integrable Functions on the Line
  • § 17. Maximal Ideals of the Rings V and V +
  • § 18. The Ring of Absolutely Integrable Functions With a Weight
  • § 19. Discrete Analogues to the Rings of Absolutely Integrable Functions
  • CHAPTER IV HARMONIC ANALYSIS ON COMMUTATIVE LOCALLY COMPACT GROUPS
  • § 20. The Group Ring of a Commutative Locally Compact Group
  • § 21. Maximal Ideals of the Group Ring and the Characters of a Group
  • § 22. The Uniqueness Theorem for the Fourier Transform and the Abundance of the Set of Characters
  • § 23. The Group of Characters
  • § 24. The Invariant Integral on the Group of Characters
  • § 25. Inversion Formulas for the Fourier Transform
  • § 26. The Pontrjagin Duality Law
  • § 27. Positive-Definite Functions
  • CHAPTER V THE RING OF FUNCTIONS OF BOUNDED VARIATION ON A LINE
  • § 28. Functions of Bounded Variation on a Line
  • § 29. The Ring of Jump Functions
  • § 30. Absolutely Continuous and Discrete Maximal Ideals of the Ring V(b)
  • § 31. Singular Maximal Ideals of the Ring V(b)
  • § 32. Perfect Sets with Linearly Independent Points. The Asymmetry of the Ring V(b)
  • § 33. The General Form of Maximal Ideals of the Ring V(b)
  • PART III
  • CHAPTER VI REGULAR RINGS
  • § 34. Definitions, Examples, and Simplest Properties
  • § 35. The Local Theorem
  • § 36. Minimal Ideals
  • § 37. Primary Ideals
  • § 38. Locally Isomorphic Rings
  • § 39. Connection between the Residue-Class Rings of Two Rings of Functions, One Embedded in the Other
  • § 40. Wiener's Tauberian Theorem
  • § 41. Primary Ideals in Homogeneous Rings of Functions
  • § 42. Remarks on Arbitrary Closed Ideals. An Example of L. Schwartz
  • CHAPTER VII RINGS WITH UNIFORM CONVERGENCE
  • § 43. Symmetric Subrings of C ( S) and Compact Extensions of a Space S
  • § 44. The Problem of Arbitrary Closed Subrings of the Ring C(S)
  • § 45. Ideals in Rings with Uniform Convergence
  • CHAPTER VIII NORMED RINGS WITH AN INVOLUTION AND THEIR REPRESENTATIONS
  • § 46. Rings with an Involution and their Representations
  • § 47. Positive Functionals and their Connection with Representations of Rings
  • § 48. Embedding of a Ring with an Involution in a Ring of Operators
  • § 49. Indecomposable Functionals and Irreducible Representations
  • § 50. The Case of Commutative Rings
  • § 51. Group Rings
  • § 52. Example of an Unsymmetric Group Ring
  • CHAPTER IX THE DECOMPOSITlON OF A COMMUTATIVE NORMED RINGINTO A DIRECT SUM OF IDEALS
  • § 53. Introduction
  • § 54. Characterization of the Space of Maximal Ideals of a Commutative Normed Ring
  • § 55. A Problem on Analytic Functions in a Finitely Generated Ring
  • § 56. Construction of a Special Finitely Generated Subring
  • § 57. Proof of the Theorem on the Decomposition of a Ring into a Direct Sum of Ideals
  • § 58. Some Corollaries
  • HISTORICO-BIBLIOGRAPHICAL NOTES
  • HISTORICO-BIBLIOGRAPHICAL NOTES
  • BIBLIOGRAPHY
  • BIBLIOGRAPHY
  • BIBLIOGRAPHY FOR APPENDIX
  • BIBLIOGRAPHY TO SECOND APPENDIX
  • INDEX
  • INDEX
  • Back Cover
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.