# [Banach] Abstract of a paper by Gilles Pisier

Dale Alspach alspach at www.math.okstate.edu
Tue Mar 16 11:55:51 CST 2004

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)
stored in gzipped form as 0403220.gz with size 26kb. The corresponding
postcript file has gzipped size 109kb.

Submitted from: gip at ccr.jussieu.fr

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http://front.math.ucdavis.edu/math.OA/0403220

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http://arXiv.org/abs/math.OA/0403220

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