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 theorems for 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, r egral and absolutely summing mappings are discussed. In what is perhaps 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 always locally reflexive. In the final section of the book, the authors consider applications to non-commutative harmonic analysis and non-self-adjoint operator algebra theory.

Table of Contents

1. Matrix and operator conventions; 2. The representation theorem; 3. Constructions and examples; 4. The extension theorem; 5. Operator systems and decompositions; 6. Injectivity; 7. The projective tensor product; 8. The injective tensor product; 9. The Haagerup tensor product; 10. Infinite matrices and asymptotic constructions; 11. The approximation property; 12. Mapping spaces; 13. Absolutely summing mappings; 14. Local reflexivity, exactness and nuclearity; 15. Local reflexivity and exact integrality; 16. Non-commutative harmonic analysis; 17. An abstract characterization for non-self-adjoint operator algebras; Appendix