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Book DetailsFields Institute MonographsVolume: 8; 1997; 165 ppMSC: Primary 46; Secondary 16; 47; 41
The nature of \(C^*\)algebras is such that one cannot study perturbation without also studying the theory of lifting and the theory of extensions. Approximation questions involving representations of relations in matrices and \(C^*\)algebras are the central focus of this volume. A variety of approximation techniques are unified by translating them into lifting problems: from classical questions about transitivity of algebras of operators on Hilbert spaces to recent results in linear algebra. One chapter is devoted to Lin's theorem on approximating almost normal matrices by normal matrices.
The techniques of universal algebra are applied to the category of \(C^*\)algebras. An important difference, central to this book, is that one can consider approximate representations of relations and approximately commuting diagrams. Moreover, the highly algebraic approach does not exclude applications to very geometric \(C^*\)algebras.
\(K\)theory is avoided, but universal properties and stability properties of specific \(C^*\)algebras that have applications to \(K\)theory are considered. Index theory arises naturally, and very concretely, as an obstruction to stability for almost commuting matrices.
Multiplier algebras are studied in detail, both in the setting of rings and of \(C^*\)algebras. Recent results about extensions of \(C^*\)algebras are discussed, including a result linking amalgamated products with the Busby/Hochshild theory.
Titles in this series are copublished with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).
Readership 
Table of Contents

Chapters

Chapter 0. Introduction

Part I. Rings and $C^*$algebras

Chapter 1. $\sigma $unital $C*$algebras

Chapter 2. Order and factoring

Chapter 3. Generators and relations

Chapter 4. Basic perturbation

Chapter 5. Pushouts and pullbacks

Chapter 6. Matrix algebras

Chapter 7. Multipliers

Part II. Lifting

Chapter 8. Easy lifting

Chapter 9. Multiplier algebras

Chapter 10. Projectivity

Chapter 11. Properties of projective $C^*$algebras

Chapter 12. Heavy lifting

Part III. Perturbing

Chapter 13. Inductive limits

Chapter 14. Stable relations

Chapter 15. Applications

Chapter 16. Extensions

Part IV. Almost Normal

Chapter 17. Normals, limits

Chapter 18. Almost normal elements

Chapter 19. Almost normal matrices


Reviews

Loring deals, in a beautifully organised and systematic manner, with a wide swathe of modern \(C^*\)algebra theory, covering such topics as generators and relations, multipliers and corona algebras, extendibility, lifting, projectivity and semiprojectivity. There is a marked algebraic flavour to much of the book, and Loring has been careful to separate out those sections that are purely ringtheoretic in nature. Algebraists are beginning to discover some of the rich structure that nonunital rings can possess, and they should find much of interest here. Loring has done a superb job in assembling a mass of powerful machinery ... Every operator algebraist will want a copy of this book.
Bulletin of the London Mathematical Society


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The nature of \(C^*\)algebras is such that one cannot study perturbation without also studying the theory of lifting and the theory of extensions. Approximation questions involving representations of relations in matrices and \(C^*\)algebras are the central focus of this volume. A variety of approximation techniques are unified by translating them into lifting problems: from classical questions about transitivity of algebras of operators on Hilbert spaces to recent results in linear algebra. One chapter is devoted to Lin's theorem on approximating almost normal matrices by normal matrices.
The techniques of universal algebra are applied to the category of \(C^*\)algebras. An important difference, central to this book, is that one can consider approximate representations of relations and approximately commuting diagrams. Moreover, the highly algebraic approach does not exclude applications to very geometric \(C^*\)algebras.
\(K\)theory is avoided, but universal properties and stability properties of specific \(C^*\)algebras that have applications to \(K\)theory are considered. Index theory arises naturally, and very concretely, as an obstruction to stability for almost commuting matrices.
Multiplier algebras are studied in detail, both in the setting of rings and of \(C^*\)algebras. Recent results about extensions of \(C^*\)algebras are discussed, including a result linking amalgamated products with the Busby/Hochshild theory.
Titles in this series are copublished with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

Chapters

Chapter 0. Introduction

Part I. Rings and $C^*$algebras

Chapter 1. $\sigma $unital $C*$algebras

Chapter 2. Order and factoring

Chapter 3. Generators and relations

Chapter 4. Basic perturbation

Chapter 5. Pushouts and pullbacks

Chapter 6. Matrix algebras

Chapter 7. Multipliers

Part II. Lifting

Chapter 8. Easy lifting

Chapter 9. Multiplier algebras

Chapter 10. Projectivity

Chapter 11. Properties of projective $C^*$algebras

Chapter 12. Heavy lifting

Part III. Perturbing

Chapter 13. Inductive limits

Chapter 14. Stable relations

Chapter 15. Applications

Chapter 16. Extensions

Part IV. Almost Normal

Chapter 17. Normals, limits

Chapter 18. Almost normal elements

Chapter 19. Almost normal matrices

Loring deals, in a beautifully organised and systematic manner, with a wide swathe of modern \(C^*\)algebra theory, covering such topics as generators and relations, multipliers and corona algebras, extendibility, lifting, projectivity and semiprojectivity. There is a marked algebraic flavour to much of the book, and Loring has been careful to separate out those sections that are purely ringtheoretic in nature. Algebraists are beginning to discover some of the rich structure that nonunital rings can possess, and they should find much of interest here. Loring has done a superb job in assembling a mass of powerful machinery ... Every operator algebraist will want a copy of this book.
Bulletin of the London Mathematical Society