Banach spaces with a weak cotype 2 property

We study the Banach spacesX with the following property: there is a numberδ in ]0,1[ such that for some constantC, any finite dimensional subspaceE ⊂X contains a subspaceF ⊂E with dimF≧δ dimE which isC-isomorphic to a Euclidean space. We show that if this holds for someδ in ]0,1[ then it also holds for allδ in ]0,1[ and we estimate the functionC=C(δ). We show that this property holds iff the “volume ratio” of the finite dimensional subspaces ofX are uniformly bounded. We also show that (althoughX can have this property without being of cotype 2)L ₂(X) possesses this property iffX if of cotype 2. In the last part of the paper, we study theK-convex spaces which have a dual with the above property and we relate it to a certain extension property.     

A construction of ℒ∞-spaces and related Banach spacesand related Banach spaces

Let λ>1. We prove that every separable Banach space E can be embedded isometrically into a separable ℒ _∞ ^λ -spaceX such thatX/E has the RNP and the Schur property. This generalizes a result in [2]. Various choices ofE allow us to answer several questions raised in the literature. In particular, takingE = ℓ₂, we obtain a ℒ _∞ ^λ -spaceX with the RNP such that the projective tensor product containsc ₀ and hence fails the RNP. TakingE=L ¹, we obtain a ℒ _∞ ^λ -space failing the RNP but nevertheless not containingc ₀.     

Complex interpolation and regular operators between Banach lattices

Supported in part by N.S.F. grant DMS 9003550.     

Non-Commutative Martingale Inequalities

We prove the analogue of the classical Burkholder-Gundy inequalites for non-commutative 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 BMO. Received: 20 March 1997 / Accepted: 21 March 1997     

Grothendieck’s theorem for operator spaces

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=T _r +T _c where T _r (resp. T _c ) 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. Oblatum 31-I-2002 & 3-IV-2002?Published online: 17 June 2002     

Bilinear forms on exact operator spaces andB(H)⊗B(H)

LetE, 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 setOS _n 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 $$B(H) \otimes _{\min } B(H) \ne B(H) \otimes _{\max } B(H),$$ 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.