Aron–Berner extensions of almost Dunford–Pettis multilinear operators

We study when Aron–Berner extensions of (separately) almost Dunford–Pettis multilinear operators between Banach lattices are (separately) almost Dunford–Pettis. For instance, for a \(\sigma \) $\sigma$ -Dedekind complete Banach lattice F containing a copy of \(\ell _\infty \) ${\ell }_{\infty }$ , we characterize the Banach lattices \(E_1, \ldots , E_m\) $E_1,\dots ,E_m$ for which every continuous m-linear operator from \(E_1 \times \cdots \times E_m\) $E_1\times \cdots \times E_m$ to F admits an almost Dunford–Pettis Aron–Berner extension. Illustrative examples are provided.