Introduction to Operator Space Theory

The first part of this book is an introduction with emphasis on examples that illustrate the theory of operator spaces. The second part is devoted to applications to C*-algebras, with a systematic exposition of tensor products of C* algebras. The third part of the book describes applications to non self-adjoint operator algebras and similarity problems. The author's counterexample to the "Halmos problem" is presented, along with work on the new concept of "length" of an operator algebra.

The theory of operator spaces is very recent and can be described as a non-commutative Banach space theory. An 'operator space' is simply a Banach space with an embedding into the space B(H) of all bounded operators on a Hilbert space H. The first part of this book is an introduction with emphasis on examples that illustrate various aspects of the theory. The second part is devoted to applications to C*-algebras, with a systematic exposition of tensor products of C*-algebras. The third (and shorter) part of the book describes applications to non self-adjoint operator algebras, and similarity problems. In particular the author's counterexample to the 'Halmos problem' is presented, as well as work on the new concept of 'length' of an operator algebra. Graduate students and professional mathematicians interested in functional analysis, operator algebras and theoretical physics will find that this book has much to offer.

Part I. Introduction to Operator Spaces: 1. Completely bounded maps; 2. Minimal tensor product; 3. Minimal and maximal operator space structures on a Banach space; 4. Projective tensor product; 5. The Haagerup tensor product; 6. Characterizations of operator algebras; 7. The operator Hilbert space; 8. Group C*-algebras; 9. Examples and comments; 10. Comparisons; Part II. Operator Spaces and C*-tensor products: 11. C*-norms on tensor products; 12. Nuclearity and approximation properties; 13. C*; 14. Kirchberg's theorem on decomposable maps; 15. The weak expectation property; 16. The local lifting property; 17. Exactness; 18. Local reflexivity; 19. Grothendieck's theorem for operator spaces; 20. Estimating the norms of sums of unitaries; 21. Local theory of operator spaces; 22. B(H) * B(H); 23. Completely isomorphic C*-algebras; 24. Injective and projective operator spaces; Part III. Operator Spaces and Non Self-Adjoint Operator Algebras: 25. Maximal tensor products and free products of non self-adjoint operator algebras; 26. The Blechter-Paulsen factorization; 27. Similarity problems; 28. The Sz-nagy-halmos similarity problem; Solutions to the exercises; References.

An introduction to the theory of operator spaces, emphasising examples that illustrate the theory and applications to C*-algebras, and applications to non self-adjoint operator algebras, and similarity problems. Postgraduate and professional mathematicians interested in functional analysis, operator algebras and theoretical physics will find the book has much to offer.