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Lifting Solutions to Perturbing Problems in \(C*\)-Algebras
Terry A. Loring, University of New Mexico, Albuquerque, NM
A co-publication of the AMS and Fields Institute.
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Fields Institute Monographs
1997; 165 pp; hardcover
Volume: 8
Reprint/Revision History:
reprinted with corrections 1998
ISBN-10: 0-8218-0602-5
ISBN-13: 978-0-8218-0602-9
List Price: US$53
Member Price: US$42.40
Order Code: FIM/8
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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 co-published with The Fields Institute for Research in Mathematical Sciences (Toronto, Ontario, Canada).

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

Table of Contents

  • Introduction
Part I. Rings and \(C^*\)-algebras
  • \(\sigma\)-unital \(C*\)-algebras
  • Order and factoring
  • Generators and relations
  • Basic perturbation
  • Push-outs and pull-backs
  • Matrix algebras
  • Multipliers
Part II.Lifting
  • Easy lifting
  • Multiplier algebras
  • Projectivity
  • Properties of projective \(C^*\)-algebras
  • Heavy lifting
Part III. Perturbing
  • Inductive limits
  • Stable relations
  • Applications
  • Extensions
Part IV. Almost Normal
  • Normals, limits
  • Almost normal elements
  • Almost normal matrices
  • Bibliography
  • Index
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