Grothendieck's Theorem for Operator Spaces

Abstract

We prove several versions of Grothendieck's Theorem for completely bounded linear maps $T:E\to F^{\ast }$, when E and F are operator spaces. We prove that if E,F are $C^{\ast }$-algebras, of which at least one is exact, then every completely bounded $T:E\to F^{\ast }$ 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 $C^{\ast }$-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 $C^{\ast }$-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 $E^{\ast }$ are exact. We also characterize the Schur multipliers which are completely bounded from the space of compact operators to the trace class.

More results and an additional section on Schur multipliers have been included

References in corpus (2)

• Noncommutative vector valued $L_p$-spaces and completely $p$-summing maps
• A simple proof of a theorem of Kirchberg and related results on $C^{\ast }$-norms

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