**Fields Institute Monographs**

Volume: 8;
1997;
165 pp;
Hardcover

MSC: Primary 46;
Secondary 16; 47; 41

Print ISBN: 978-0-8218-0602-9

Product Code: FIM/8

List Price: $56.00

Individual Member Price: $44.80

**Electronic ISBN: 978-1-4704-3135-8
Product Code: FIM/8.E**

List Price: $56.00

Individual Member Price: $44.80

# Lifting Solutions to Perturbing Problems in $C*$-Algebras

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*Terry A. Loring*

A co-publication of the AMS and Fields Institute

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 co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

#### Table of Contents

# Table of Contents

## Lifting Solutions to Perturbing Problems in $C*$-Algebras

- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Introduction 112
- Part I. Rings and 𝐶*-algebras 1122
- 𝜎-unital 𝐶*-algebras 1324
- Order and factoring 1930
- Generators and relations 2334
- Basic perturbation 3142
- Push-outs and pull-backs 3748
- Matrix algebras 4152
- Multipliers 4758
- Part II.Lifting 5970
- Easy lifting 6172
- Multiplier algebras 6778
- Projectivity 7384
- Properties of projective 𝐶*-algebras 8192
- Heavy lifting 89100
- Part III. Perturbing 97108
- Inductive limits 99110
- Stable relations 105116
- Applications 113124
- Extensions 121132
- Part IV. Almost Normal 129140
- Normals, limits 131142
- Almost normal elements 137148
- Almost normal matrices 147158
- Bibliography 157168
- Index 165176
- Back Cover Back Cover1177

#### Readership

#### 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 ring-theoretic 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