Completely bounded maps into certain Hilbertian operator spaces
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We prove a factorization of completely bounded maps from a $C^*$-algebra $A$ (or an exact operator space $Esubset A$) to $ell_2$ equipped with the operator space structure of $(C,R)_ heta$ ($0< heta<1$) obtained by complex interpolation between the column and row Hilbert spaces. More precisely, if $F$ denotes $ell_2$ equipped with the operator space structure of $(C,R)_ heta$, then $u: A o F$ is completely bounded iff there are states $f,g$ on $A$ and $C>0$ such that [ forall ain Aquad |ua|^2le C f(a^*a)^{1- heta}g(aa^*)^{ heta}.] This extends the case $ heta=1/2$ treated in a recent paper with Shlyakhtenko. The constants we obtain tend to 1 when $ heta o 0$ or $ heta o 1$. We use analogues of "free Gaussian" families in non semifinite von Neumann algebras. As an application, we obtain that, if $0< heta<1$, $(C,R)_ heta$ does not embed completely isomorphically into the predual of a semifinite von Neumann algebra. Moreover, we characterize the subspaces $Ssubset Roplus C$ such that the dual operator space $S^*$ embeds (completely isomorphically) into $M_*$ for some semifinite von neumann algebra $M$: the only possibilities are $S=R$, $S=C$, $S=Rcap C$ and direct sums built out of these three spaces. We also discuss when $Ssubset Roplus C$ 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.