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

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 $B({\mathrm{\ell }}_{\mathrm{\infty }},X^{\ast })={\mathrm{\Pi }}_2({\mathrm{\ell }}_{\mathrm{\infty }},X^{\ast })$. 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 ${\mathrm{\ell }}_p$-spaces, $1\le p<\mathrm{\infty }$.