Completely bounded maps into certain Hilbertian operator spaces
| dc.creator | Pisier, Gilles | |
| dc.date | 2004-03-13 | |
| dc.date | 2004-07-07 | |
| dc.date.accessioned | 2026-07-07T05:06:22Z | |
| dc.date.available | 2026-07-07T05:06:22Z | |
| dc.description | 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 $(C,R)_θ$ ($0<θ<1$) 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 $(C,R)_θ$, then $u: A \to F$ is completely bounded iff there are states $f,g$ on $A$ and $C>0$ such that \[ \forall a\in A\quad \|ua\|^2\le C f(a^*a)^{1-θ}g(aa^*)^θ.\] This extends the case $θ=1/2$ treated in a recent paper with Shlyakhtenko. The constants we obtain tend to 1 when $θ\to 0$ or $θ\to 1$. We use analogues of "free Gaussian" families in non semifinite 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 $S\subset R\oplus C$ 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. | |
| dc.description | Minor corrections of misprints and addition of an introduction | |
| dc.identifier | https://arxiv.org/abs/math/0403220 | |
| dc.identifier | http://arxiv.org/abs/math/0403220 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/70445 | |
| dc.subject | Operator Algebras | |
| dc.subject | Functional Analysis | |
| dc.subject | 46L07, 46L54, 47L25, 47L50 | |
| dc.title | Completely bounded maps into certain Hilbertian operator spaces | |
| dc.type | text |