Complemented copies of \(l_ 1\) in \(L^\infty(\mu,X)\) (Q1384867)

An active field of research in recent years has been the study of the inclusion, as a subspace or complemented subspace, of classical Banach sequence spaces such as \(c_0\), \(\ell_1\) or \(\ell_\infty\) in Banach spaces \(L^p(\mu, B)\) of Bochner \(p\)-integrable (essentially bounded for \(p=\infty\)) functions over a finite measure space \((\Omega,\Sigma,\mu)\) with values in a Banach space \(X\). The following problem was originally posed by Labuda: When does \(L^\infty(\mu, X)\) contain a complemented copy of \(\ell_1\)? Natural conjectures such as ``if (and only if) \(X\) has a (complemented) copy of \(\ell_1\)'', were disproved by an example due to Montgomery-Smith: there is a Banach space \(X\) with separable dual such that \(L^\infty(\mu, X)\) contains a complemented copy of \(\ell_1\). The aim of this paper is to answer this question for the case when \(X\) is a Banach lattice.