The Maurey extension property for Banach spaces with the Gordon–Lewis property and related structures

P. G. Casazza ¹   ; N. J. Nielsen ²

¹ Department of Mathematics University of Missouri Columbia, MO 65211, U.S.A.
² Department of Mathematics and Computer Science University of Southern Denmark Campusvej 55 DK-5230 Odense M, Denmark

Studia Mathematica, Tome 155 (2003) no. 1, pp. 1-21

Résumé

The main result of this paper states that if a Banach space $X$ has the property that every bounded operator from an arbitrary subspace of $X$ into an arbitrary Banach space of cotype 2 extends to a bounded operator on $X$, then every operator from $X$ to an $L_1$-space factors through a Hilbert space, or equivalently $B(\ell _{\infty },X^*)={\mit \Pi }_2(\ell _{\infty },X^*)$. If in addition $X$ has the Gaussian average property, then it is of type 2. This implies that the same conclusion holds if $X$ has the Gordon–Lewis property (in particular $X$ could be a Banach lattice) or if $X$ is isomorphic to a subspace of a Banach lattice of finite cotype, thus solving the Maurey extension problem for these classes of spaces. The paper also contains a detailed study of the property of extending operators with values in $\ell _p$-spaces, $1\le p\infty $.

Détail

DOI : 10.4064/sm155-1-1

Keywords: main result paper states banach space has property every bounded operator arbitrary subspace arbitrary banach space cotype extends bounded operator every operator space factors through hilbert space equivalently ell infty * mit ell infty * addition has gaussian average property type implies conclusion holds has gordon lewis property particular could banach lattice isomorphic subspace banach lattice finite cotype solving maurey extension problem these classes spaces paper contains detailed study property extending operators values ell p spaces infty