# Completely bounded maps into certain Hilbertian operator spaces

Research paper by **Gilles Pisier**

Indexed on: **07 Jul '04**Published on: **07 Jul '04**Published in: **Mathematics - Operator Algebras**

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### Abstract

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