On extension of positive multilinear operators

Using the linearization of positive multilinear operators by means of the Fremlin tensor product of vector lattices one can show that a multilinear operator acting from the Cartesian product of majorizing subspaces of vector lattices to Dedekind complete vector lattice admits an extension to a positive multilinear operator defined on the Cartesian product of the ambient vector lattices. In this note, we establish that this result remains valid if the multilinear operator is defined on the Cartesian product of majorizing subspaces of separable Banach lattices and takes values in a topological vector lattice with the $\sigma$-interpolation property, provided that the mentioned Banach lattices have the property of subadditivity. The latter ensures that the algebraic tensor product of the majorizing subspaces is majorizing in the Fremlin tensor product of the Banach lattices under consideration. An open question is stated: whether or not the result remains valid if the subadditivity property is omitted (or weakened). The possibility of weakening the requirement of order completeness of the target lattice by imposing some additional requirements on the domain vector lattices was first observed by Abramovich and Wikstead when proving one version of the Hahn–Banach–Kantorovich theorem.