Grothendieck's Theorem for Operator Spaces

Abstract

We prove several versions of Grothendieck's Theorem for completely bounded linear maps $T\colon E \to F^*$, 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 $T\colon E \to F^*$ 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.