This is an announcement for the paper "Stabilizing isomorphisms from $\ell_p(\ell_2)$ into $L_p[0,1]$" by Ran Levy and Gideon Schechtman.
Abstract: Let $1<p\not=2<\infty$, $\epsilon>0$ and let $T:\ell_p(\ell_2)\overset{into}{\rightarrow}L_p[0,1]$ be an isomorphism. Then there is a subspace $Y\subset \ell_p(\ell_2)$ $(1+\epsilon)$-isomorphic to $\ell_p(\ell_2)$ such that: $T_{|Y}$ is an $(1+\epsilon)$-isomorphism and $T\left(Y\right)$ is $K_p$-complemented in $L_p[0,1]$, with $K_p$ depending only on $p$. Moreover, $K_p\le (1+\epsilon)\gamma_p$ if $p>2$ and $K_p\le (1+\epsilon)\gamma_{p/(p-1)}$ if $1<p<2$, where $\gamma_r$ is the $L_r$ norm of a standard Gaussian variable.
Archive classification: math.FA
Mathematics Subject Classification: 46E30
Submitted from: gideon@weizmann.ac.il
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1103.0047
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