On Asplund Spaces C_k(X) and w^∗ -Binormality

A celebrated theorem of Namioka and Phelps (Duke Math J 42:735–750, 1975) says that for a compact space X, the Banach space C(X) is Asplund iff X is scattered. In our paper we extend this result to the space of continuous real-valued functions endowed with the compact-open topology C_k(X) for several natural classes of non-compact Tychonoff spaces X. The concept of Δ ₁ -spaces introduced recently in Ka̧kol et al. (Some classes of topological spaces extending the class of Δ -spaces, submitted for publication) has been shown to be applicable for this research. w^∗ -binormality of the dual of the Banach space C(X) implies that C(X) is Asplund (Kurka in J Math Anal Appl 371:425–435, 2010). In our paper we prove in particular that for a Corson compact space X the converse is true. We establish a tight relationship between the property of w^∗ -binormality of the dual C(X) ^′ and the class of compact Δ -spaces X introduced and explored earlier in Ka̧kol and Leiderman (Proc Am Math Soc Ser B 8:86–99, 2021, 8:267–280, 2021). We find a complete characterization of a compact space X such that the dual C(X) ^′ possesses a stronger property called effective w^∗ -binormality. We provide several illustrating examples and pose open questions.