  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, $1\le p<\infty$.
