Gilles Pisier

Martingales with values in uniformly convex spaces

Using the techniques of martingale inequalities in the case of Banach space valued martingales, we give a new proof of a theorem of Enflo: every super-reflexive space admits an equivalent uniformly convex norm. Letr be a number in ]2, ∞[; we prove moreover that if a Banach spaceX is uniformly convex (resp. ifδx(ɛ)/ɛr whenɛ → 0) thenX admits for someq<∞ (resp. for someq<r) an equivalent norm for which the corresponding modulus of convexity satisfiesδ(ɛ)/ɛq → ∞ whenɛ → 0. These results have dual analogues concerning the modulus of smoothness. Our method is to study some inequalities for martingales with values in super-reflexive or uniformly convex spaces which are characteristic of the geometry of these spaces up to isomorphism.

Characterizations of almost surely continuous p-stable random Fourier series and strongly stationary processes

On etend le probleme de Dudley-Fernique aux processus p-stables fortement stationnaires, 1

Non commutative Khintchine and Paley inequalities

considered as a norm on the set of all finitely supported sequences (x,,) in X. While a satisfactory solution seems hopeless at the moment for an arbitrary space X, there are cases for which the answer is known to be very simple and as complete as possible. For instance, if X is the Banach space Lp(Q, ~ , m) (1 ~ p < ~ ) the classical Khintchine inequalities (cf. [LT, I.d.6]) and Fubinis theorem imply that there is an absolute constant C such that, for all x,, in X=Lv(.Q, ~ , m), we have

Grothendieck's theorem for noncommutative C∗-algebras, with an Appendix on Grothendieck's constants

Abstract We study a conjecture of Grothendieck on bilinear forms on a C∗-algebra Ol . We prove that every “approximable” operator from Ol into Ol ∗ factors through a Hilbert space, and we describe the factorization. In the commutative case, this is known as Grothendiecks theorem. These results enable us to prove a conjecture of Ringrose on operators on a C∗-algebra. In the Appendix, we present a new proof of Grothendiecks inequality which gives an improved upper bound for the so-called Grothendieck constant kG.

Bilinear Forms on Exact Operator Spaces and B(H) ⊗ B(H)

AbstractLetE, F be exact operator spaces (for example subspaces of theC*-algebraK(H) of all the compact operators on an infinite dimensional Hilbert spaceH). We study a class of bounded linear mapsu: E →F* which we call tracially bounded. In particular, we prove that every completely bounded (in shortc.b.) mapu: E →F* factors boundedly through a Hilbert space. This is used to show that the setOSn of alln-dimensional operator spaces equipped with thec.b. version of the Banach Mazur distance is not separable ifn>2.As an application we whow that there is more than oneC*-norm onB (H) ⊗ B (H), or equivalently that

Banach spaces with a weak cotype 2 property

B(H) \otimes _{\min } B(H) \ne B(H) \otimes _{\max } B(H),

The strongp-variation of martingales and orthogonal series

which answers a long standing open question. Finally we show that every “maximal” operator space (in the sense of Blecher-Paulsen) is not exact in the infinite dimensional case, and in the finite dimensional case, we give a lower bound for the “exactness constant”. In the final section, we introduce and study a new tensor product forC*-albegras and for operator spaces, closely related to the preceding results.

Complex interpolation and regular operators between Banach lattices

Abstract.We prove several versions of Grothendieck’s Theorem for completely bounded linear maps T:E→F*, when E and F are operator spaces. We prove that if E, F are C*-algebras, of which at least one is exact, then every completely bounded T:E→F* can be factorized through the direct sum of the row and column Hilbert operator spaces. Equivalently T can be decomposed as T=Tr+Tc where Tr (resp. Tc) factors completely boundedly through a row (resp. column) Hilbert operator space. This settles positively (at least partially) some earlier conjectures of Effros-Ruan and Blecher on the factorization of completely bounded bilinear forms on C*-algebras. Moreover, our result holds more generally for any pair E, F of “exact” operator spaces. This yields a characterization of the completely bounded maps from a C*-algebra (or from an exact operator space) to the operator Hilbert space OH. As a corollary we prove that, up to a complete isomorphism, the row and column Hilbert operator spaces and their direct sums are the only operator spaces E such that both E and its dual E* are exact. We also characterize the Schur multipliers which are completely bounded from the space of compact operators to the trace class.