Introduction to Operator Space Theory

An introduction to the theory of operator spaces, emphasising applications to C*-algebras.

Gilles Pisier (Author)

9780521811651, Cambridge University Press

Paperback, published 25 August 2003

488 pages
22.9 x 15.3 x 2.6 cm, 0.645 kg

'The tone of the book is quite informal, friendly and inviting. Even to experts in the field, a large proportion of the results, and certainly of the proofs, will be new and stimulating. … there are literally thousands of wonderful results and insights in the text which the reader will not find elsewhere. The book covers an incredible amount of ground, and makes use of some of the most exciting recent work in modern analysis. … It is a magnificent book: an enormous treasure trove, and a work of love and care by one of the great analysts of our time. All students and researchers in functional analysis should have a copy. Anybody planning to work in operator space theory will need to be thoroughly immersed in it.' Proceedings of the Edinburgh Mathematical Society

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.

Subject Areas: Functional analysis & transforms [PBKF]