Grothendieck’s theorem for operator spaces

By Gilles Pisier, D. Shlyakhtenko. 2001

106 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/60301bbc50150b617bf0596ca7c2b6b513757586

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Citations in the corpus

Embedding of the operator space OH and the logarithmic ‘little Grothendieck inequality’ (2003, background): “conditional expectation from N = ¯k≤l(Tλk/λl , φλk/λl ) ⊗ Mn onto M. According t o [PS, Lemma 2.5], the von Neumann algebra ¯k≤l(Tλk/λl , φαk/λl ) is QWEP and the assertion follows.”
Grothendieck's Theorem, past and present (2011, methodology): “A posteriori, this means (again!) that the approximability assumption of [132] for c.”
Operator spaces with few completely bounded maps (2004, methodology): “Remark 4 Instead of [18], we could have used the stronger results about the non commutative Grothendieck inequality of [ 36 ].”
A reduction method for noncommutative L_p-spaces and applications (2008, methodology): “This illustration is further shown in the works [PS] and [X4] on operator space Grothendieck theorems.”
Noncommutative Burkholder/Rosenthal inequalities II: Applications (2007, methodology): “for all xk 2 Lq(Ak) with mean 0. Our aim is to prove the estimate for p. Of course we may assume p > 2. Let xk 2 Lp(Ak) and EM (xk) = 0. First we apply the Khintchine inequality (see [ PS ] for the right order of constants) and deduce from Lemma 1.1 that”
Rosenthal Type inequalities for free chaos. (2005, background): “ circular system in Voiculescu’s sense 44 JUNGE, PARCET AND XU and the result below reduces to Theorem 8.6.5 in [31]. On the other hand, the case p= ∞ was already proved by Pisier and Shlyakhtenko in [33]. Theorem5.1. Let N be a von Neumann algebra and 1 ≤ p≤ ∞. Let us consider a ﬁnite sequence x1,x”