Introduction to Operator Space Theory

Lecture notes series 294

Gilles Pisier

478 pages, parution le 21/08/2003

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Résumé

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
- Completely bounded maps
- Minimal tensor product
- Minimal and maximal operator space structures on a Banach space
- Projective tensor product
- The Haagerup tensor product
- Characterizations of operator algebras
- The operator Hilbert space
- Group C*-algebras
- Examples and comments
- Comparisons
Part II. Operator Spaces and C*-tensor products
- C*-norms on tensor products
- Nuclearity and approximation properties
- C*
- Kirchberg's theorem on decomposable maps
- The weak expectation property
- The local lifting property
- Exactness
- Local reflexivity
- Grothendieck's theorem for operator spaces
- Estimating the norms of sums of unitaries
- Local theory of operator spaces
- B(H) * B(H)
- Completely isomorphic C*-algebras
- Injective and projective operator spaces
Part III. Operator Spaces and Non Self-Adjoint Operator Algebras
- Maximal tensor products and free products of non self-adjoint operator algebras
- The Blechter-Paulsen factorization
- Similarity problems
- The Sz-nagy-halmos similarity problem
- Solutions to the exercises
- References

Caractéristiques techniques

	PAPIER
Éditeur(s)	Cambridge University Press
Auteur(s)	Gilles Pisier
Parution	21/08/2003
Nb. de pages	478
Format	15 x 23
Couverture	Broché
Poids	640g
Intérieur	Noir et Blanc
EAN13	9780521811651
ISBN13	978-0-521-81165-1