refinement of a lemma by Aron, Cascales, and Kozhushkina on Asplund operators

We prove a refinement of a lemma by Aron, Cascales, and Kozhushkina on Asplund operators. Refinements of a Bishop–Phelps–Bollobás type theorem for Asplund operators with values in spaces C0(L) by the same authors, and an extension of this theorem for Asplund operators with values in uniform algebras by Cascales, Guirao, and Kadets, follow.