[Banach] Abstract of a paper by Gilles Pisier

[Dale Alspach]

This is an announcement for the paper "Completely bounded maps into
certain Hilbertian operator spaces" by Gilles Pisier.


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"), which
somewhat unifies Junge's ideas with those of our work with Shlyakhtenko.

Archive classification: Operator Algebras; Functional Analysis

Mathematics Subject Classification: 46L07, 46L54, 47L25, 47L50

The source file(s), oh3.suite.09.mars.04.tex: 79910 bytes, is(are)
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Submitted from: gip at ccr.jussieu.fr

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