Operator Spaces (London Mathematical Society Monographs, 23) by Edward G. Effros (PDF)

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Operator space theory provides a synthesis of Banach space theory with the non-commuting ‘quantum’ variables of operator algebra theory, and it has led to exciting new approaches in both disciplines. The authors begin by giving completely elementary proofs of the basic representation theoremsfor abstract operator spaces and their mappings. This is followed by a discussion of tensor products and the analogue of Grothendieck’s approximation property. In the next section, the operator space analogues of the nuclear, integral and absolutely summing mappings are discussed. In what isperhaps the deepest part of the book, the authors present the remarkable ‘non-classical’ phenomena that occur when one considers local reflexivity and exactness for operator spaces. They have included the recent proof that, in contrast to C*-algebras themselves, C*-algebraic duals are alwayslocally reflexive. In the final section of the book, the authors consider applications to non-commutative harmonic analysis and non-self-adjoint operator algebra theory.

User’s Reviews

Editorial Reviews: Review “[A] very good modern introduction to the subject… very well suited for an introductory course. Most of the basic tions are treated in an elegant way and the proofs of the main results are accessible to the non-expert.”–Mathematical Abstracts”The authors define an (abstract) operator space by axiomatising the properties of concrete operator spaces. . .The authors claim that their ‘goal in this monograph has been to explain the deep analogy between linear spaces of bounded functions and linear spaces of bounded functions and linearspaces of bounded operators’ and that the ‘operator space theory will provide Banach space theorists with exciting new vistas for research’. The monograph is designed for graduate students and researchers interested in the filed, and can be understood with a rudimentary knowledge of functionalanalysis. and in particular of Banach space theory.”–EMS About the Author Professor E G Effros Department of Mathematics University of California LOS ANGELES, CA 90024-1555 USA Professor Z-J Ruan Department of Mathematics University of Illinois Urbana, IL 61801 USA

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