Completely bounded maps into certain Hilbertian operator spaces

Abstract

We prove a factorization of completely bounded maps from a C*-algebra A (or an exact operator space E ⊂ A) to ℓ2 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 ℓ2 equipped with the operator space structure of (C,R)θ, then u: A → F is completely bounded if and only if there are states f, g on A and C > 0 such that for all a ∈ A, ‖ua‖2≤Cf(a*a)1−θg(aa*)θ⁠. This extends the case θ = 1/2 treated in a recent paper with Shlyakhtenko (2002). 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 analogs of “free Gaussian” families in nonsemifinite 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 noncommutative L1-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”), which somewhat unifies Junge's ideas with those of our work with Shlyakhtenko (2002).