The solution to the Maurey extension problem for Banach spaces with the Gordon Lewis property and related structures

TL;DR

This paper characterizes Banach spaces with the Gordon-Lewis property that allow bounded operator extensions, establishing new connections with cotype 2 and type 2 properties, and advances the Maurey extension problem.

Contribution

It solves the Maurey extension problem for Banach spaces with the Gordon-Lewis property and related structures, linking operator extension properties to geometric space properties.

Findings

Spaces with the Gordon-Lewis property extend operators into cotype 2 spaces.

Such spaces are of type 2 if they have the Gaussian average property.

The paper studies operator extensions into ll_p spaces for 1 st p < init.

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 $B(\ell_{\infty},X^*)=\Pi_2(\ell_{\infty},X^*)$. 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 property for these classes of spaces. The paper also contains a detailed study of the property of extending operators with values in $\ell_p$-spaces, $1\le p<\infty$.