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

### Volume 155 / 2003

#### Abstract

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 $.