Completely Bounded Maps into Certain Hilbertian Operator Spaces

We prove a factorization of completely bounded maps from a C *-algebra A (or an exact operator space E ⊂ A) to ℓ 2 equipped with the operator space structure of (C, R) θ (0 < θ < 1) obtained by complex interpolation between the column and row Hilbert spaces. More precisely, if F denotes ℓ 2 equipped with the operator space structure of (C, R) θ , then u : A → F is completely bounded iff there are states f, g on A and C > 0 such that ∀a ∈ A ua 2 ≤ Cf (a * a) 1−θ g(aa *) θ. This extends the case θ = 1/2 treated in a recent paper with Shlyakhtenko [25]. The constants we obtain tend to 1 when θ → 0 or θ → 1, so that we recover, when θ = 0 (or θ = 1), the case of mappings into C (or into R), due to Effros and Ruan. We use analogues of " free Gaussian " families in non semifinite von Neumann algebras. As an application, we obtain that, if 0 < θ < 1, (C, R) θ does not embed completely isomorphically into the predual of a semifinite von Neumann algebra. Moreover, we characterize the subspaces S ⊂ R ⊕ C such that the dual operator space S * embeds (completely isomorphically) into M * for some semifinite von neumann algebra M : the only possibilities are S = R, S = C, S = R ∩ C and direct sums built out of these three spaces. We also discuss when S ⊂ R ⊕ C is injective, and give a simpler proof of a result due to Oikhberg on this question. In the appendix, we present a proof of Junge's theorem that OH embeds completely isomorphically into a non-commutative L 1-space. The main idea is similar to Junge's, but we base the argument on complex interpolation and Shlyakhtenko's generalized circular systems (or " generalized free Gaussian "), that somewhat unifies Junge's ideas with those of our work with Shlyakhtenko [25].