Given Banach spaces E, F, G, we denote by \(\mathcal L^2_\text {I}(E,F;G)\) the space of integral bilinear mappings from \(E\times F\) into G, by \({\mathcal {L}}^2_\text {N}(E,F;G)\) the space of nuclear bilinear maps, and by \({\mathcal {L}}^k_{\ell _1,\text {I}}(E,F;G)\) the space of left integral \(\ell _1\) -factorable bilinear maps. When G is the scalar field, it is omitted. It is well-known that \({\mathcal {L}}^2_{\text{ N }}\subseteq {\mathcal {L}}^2_{\ell _1,\text {I}} \subseteq {\mathcal {L}}^2_\text {I}\) . We prove that, if the injective tensor product \(E{{\widehat{\otimes }}}_\epsilon F\) contains no copy of \(\ell _1\) , then \({\mathcal {L}}^2_\text {I}(E,F)\equiv {\mathcal {L}}^2_\text {N}(E,F)\) , where the symbol \(\equiv \) means “isometrically isomorphic". It is also shown that the latter identity is equivalent to \({\mathcal {L}}^k_{\ell _1,\text {I}}(E,F;G)\equiv {\mathcal {L}}^2_\text {N}(E,F;G)\) for every Banach space G. We find other necessary conditions for this identity in terms of isomorphic properties of E and F. All the above mentioned results are also valid for k-linear mappings \((k\ge 2)\) . The arguments used allow for identification of a large class of integral multilinear mappings which are automatically nuclear. As a consequence, we give a simple proof of a Grothendieck type theorem stating that a multilinear mapping on a product of \(L_1(\mu )\) spaces is Riesz-representable if and only if its composition with the canonical inclusions \(L_\infty (\mu )\rightarrow L_1(\mu )\) is nuclear. A more involved proof of this result had been given in 2024. It is proved that the spaces \(E_1,\ldots , E_k\) are \(\mathscr {L}_\infty \) -spaces if and only if the 1-dominated k-linear mappings from \(E_1\times \cdots \times E_k\) into an arbitrary Banach space coincide with a subclass of the k-linear integral mappings (called S-factorable). This extends to the multilinear case a result of Stegall and Retherford, and strengthens a previous partial extension of 2002.
