Part I. INTRODUCTION TO OPERATOR SPACES 1. Completely bounded maps 2. The minimal tensor product. Ruan's theorem. Basic operations 2.1. Minimal tensor product 2.2. Ruan's theorem 2.3. Dual space 2.4. Quotient space Quotient by a subspace Quotient by an ideal 2.5. Bidual. Von Neumann algebras 2.6. Direct sum 2.7. Intersection, sum, complex interpolation 2.8. Ultraproduct 2.9. Complex conjugate 2.10. Opposite 2.11. Ruan's theorem and quantization 2.12. Universal objects 2.13. Perturbation lemmas 3. Minimal and maximal operator space structures 4. Projective tensor product 5. The Haagerup tensor product Basic properties Multilinear factorization Injectivity/projectivity Self-duality Free products Factorization through R or C Symmetrized Haagerup tensor product Complex interpolation 6. Characterizations of operator algebras 7. The operator Hilbert space Hilbertian operator spaces Existence and unicity of OH. Basic properties Finite-dimensional estimates Complex interpolation Vector-valued L p-spaces, either commutative or noncommutative 8. Group C *-algebras. Universal algebras and unitization for an operator space 9. Examples and comments 9.1. A concrete quotient: Hankel matrices vi Contents 9.2. Homogeneous operator spaces 9.3. Fermions. Antisymmetric Fock space. Spin systems 9.4. The Cuntz algebra O n 9.5. The operator space structure of the classical L p-spaces 9.6. The C *-algebra of the free group with n generators 9.7. Reduced C *-algebra of the free group with n generators 9.8. Operator space generated in the usual L p-space by Gaussain random variables or by the Rademacher functions 9.9. Semi-circular systems in Voiculescu's sense 9.10. Embeddings of von Neumann algebras into ultraproducts 9.11. Dvoretzky's theorem 10. Comparisons Part II. OPERATOR SPACES AND C *-TENSOR PRODUCTS 11. C *-norms on tensor products. Decomposable maps. Nuclearity 12. Nuclearity and approximation properties 13. C * (F ∞) ⊗ B(H) 14. Kirchberg's theorem on decomposable maps 15. The Weak Expectation Property (WEP) 16. The Local Lifting Property (LLP) 17. Exactness 18. Local reflexivity Basic properties A conjecture on local reflexivity and OLLP Properties C, C , and C. Exactness versus local reflexivity 19. Grothendieck's theorem for operator spaces 20. Estimating the norms of sums of unitaries: Ramanujan graphs, property T , random matrices 21. Local theory of operator spaces. Nonseparability of OS n 22. B(H) ⊗ B(H) 23. Completely isomorphic C *-algebras 24. Injective and projective operator spaces

This book aims to give a self-contained presentation of a number of results, which relate the volume of convex bodies in n-dimensional Euclidean space and the geometry of the corresponding finite-dimensional normed spaces. The methods employ classical ideas from the theory of convex sets, probability theory, approximation theory and the local theory of Banach spaces. The book is in two parts. The first presents self-contained proofs of the quotient of the subspace theorem, the inverse Santalo inequality and the inverse Brunn-Minkowski inequality. The second part gives a detailed exposition of the recently introduced classes of Banach spaces of weak cotype 2 or weak type 2, and the intersection of the classes (weak Hilbert space). The book is based on courses given in Paris and in Texas.

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Abstract. We prove the analogue of the classical Burkholder-Gundy inequalites for noncommutative martingales. As applications we give a characterization for an Ito-Clifford integral to be an L p -martingale via its integrand, and then extend the Ito-Clifford integral theory in L 2 , developed by Barnett, Streater and Wilde, to L p for all 1 < p < ∞. We include an appendix on the non-commutative analogue of the classical Fefferman duality between H 1 and BM O.Plan. One interesting feature of our work, is that the square function is defined differently (and it must be changed!) according to p < 2 or p > 2. This surprising phenomenon was already discovered by F. Lust-Piquard in [LP] (see also [LPP]) while establishing non-commutative versions of Khintchine's inequalities.Let us briefly describe our main inequality. Let M be a finite von Neumann algebra with a normalized normal faithful trace τ , and (M n ) ≥0 be an increasing filtration of von Neumann subalgebras of M. Let 1 < p < ∞ and (x n ) be a martingale with respect toThen our main result reads as follows. If p ≥ 2, we have (with equivalence constants depending only on p)This is no longer valid for p < 2; however for p < 2 the "right" inequalities arewhere the infimum runs over all decompositions d n = a n + b n of d n as a sum of martingale difference sequences adapted to the same filtration.In particular, this applies to martingale transforms: given a martingale (x n ) as above and an adapted bounded sequence ξ = (ξ n ), i.e. such that ξ n ∈ M n for all n ≥ 0, we can form the martingaleis bounded in M and if ξ n−1 commutes with M n for all n, the transformedIndeed, by duality, it suffices to check this for p ≥ 2, and then it is an easy consequence of (0.1). Note however that the preceding statement can fail if one does not assume that ξ n−1 commutes with M n . In the case p ≥ 2, it suffices to assume that ξ n−1 commutes with x n − x n−1 for all n. The latter assumption is used to show that if, say ξ n−1 ≤ 1, we have (y n − y n−1 )(y n − y n−1 ) * ≤ (x n − x n−1 )(x n − x n−1 ) * . Of course, this assumption can be relaxed further, all that is needed is to be able to compare the "square functions" associated to (y n ) and (x n ) appearing on the right in (0.1).In section 2 the above inequalities (0.1) and (0.2) are proved. The key point of our proof is the following passage: assuming the above inequalities for some 1 < p < ∞, then we deduce them for 2p. The rest of the proof can be accomplished by iteration (starting from p = 2), interpolation and duality. We would like to emphasize that this proof is entirely self-contained.The style of proof of (0.1) and (0.2) is rather old fashioned: it is reminiscent of Marcel Riesz's classical argument for the boundedness of the Hilbert transform on L p (1 < p < ∞), and also of Paley's proof of (0.1) in the classical dyadic case ([Pa]), i.e.when M n = L ∞ ({−1, +1} n ). It has been known for many years that Marcel Riesz's argument could be easily adapted to prove the boundedness of the Hilbert transform on if, for any 1...