Local Multipliers of C*-Algebras

1. Front Matter

   Pages i-xii

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2. Introduction

   • Pere Ara, Martin Mathieu

   Pages 1-6

3. Prerequisites

   • Pere Ara, Martin Mathieu

   Pages 7-48

4. The Symmetric Algebra of Quotients and its Bounded Analogue

   • Pere Ara, Martin Mathieu

   Pages 49-70

5. The Centre of the Local Multiplier Algebra

   • Pere Ara, Martin Mathieu

   Pages 71-95

6. Automorphisms and Derivations

   • Pere Ara, Martin Mathieu

   Pages 97-168

7. Elementary Operators and Completely Bounded Mappings

   • Pere Ara, Martin Mathieu

   Pages 169-235

8. Lie Mappings and Related Operators

   • Pere Ara, Martin Mathieu

   Pages 237-295

9. Back Matter

   Pages 297-319

   PDF

Many problems in operator theory lead to the consideration ofoperator equa­ tions, either directly or via some reformulation. More often than not, how­ ever, the underlying space is too 'small' to contain solutions of these equa­ tions and thus it has to be 'enlarged' in some way. The Berberian-Quigley enlargement of a Banach space, which allows one to convert approximate into genuine eigenvectors, serves as a classical example. In the theory of operator algebras, a C*-algebra A that turns out to be small in this sense tradition­ ally is enlarged to its (universal) enveloping von Neumann algebra A". This works well since von Neumann algebras are in many respects richer and, from the Banach space point of view, A" is nothing other than the second dual space of A. Among the numerous fruitful applications of this principle is the well-known Kadison-Sakai theorem ensuring that every derivation 8 on a C*-algebra A becomes inner in A", though 8 may not be inner in A. The transition from A to A" however is not an algebraic one (and cannot be since it is well known that the property of being a von Neumann algebra cannot be described purely algebraically). Hence, ifthe C*-algebra A is small in an algebraic sense, say simple, it may be inappropriate to move on to A". In such a situation, A is typically enlarged by its multiplier algebra M(A).

• C*-algebra
• algebra
• automorphism
• operator theory
• ring

• Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, Spain

  Pere Ara

• Department of Pure Mathematics School of Mathematics and Physics, Queen’s University Belfast, Belfast, Northern Ireland

  Martin Mathieu

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