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 \(\sigma \)-Dedekind complete Banach lattice F containing a copy of \(\ell _\infty \), we characterize the Banach lattices \(E_1, \ldots , E_m\) for which every continuous m-linear operator from \(E_1 \times \cdots \times E_m\) to F admits an almost Dunford–Pettis Aron–Berner extension. Illustrative examples are provided.