Grothendieck’s theorem for operator spaces

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.

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