Strictly convex norms, \(G_\delta\)-diagonals and non-Gruenhage spaces (Q2845736)

In [\textit{J. Orihuela, R. J. Smith} and \textit{S. Troyanski}, Proc.\ Lond.\ Math.\ Soc.\ (3) 104, No.\ 1, 197--222 (2012; Zbl 1241.46005)], a topological property called \((\ast)\) was introduced to study Banach spaces which admit strictly convex norms. The class of topological spaces having \((\ast)\) includes the Gruenhage spaces and spaces having a \(G_\delta\) diagonal, and it was shown that a scattered compact space \(K\) has \((\ast)\) if and only if \(C(K)^*\) admits a strictly convex dual norm. Under additional hypotheses, the authors showed the existence of scattered compact, non-Gruenhage spaces having \((\ast)\).NEWLINENEWLINE The purpose of the article under review is to show that such examples can exist in ZFC. In particular, the author constructs a locally compact, scattered Hausdorff non-Gruenhage space \(D\) having a \(G_\delta\)-diagonal. This imples that Gruenhage spaces are not necessary for the construction of strictly convex dual norms on dual Banach spaces. Additionally, the author proves that the Banach space \(C_0(D)\) of continuous real-valued functions vanishing at infinity admits a \(C^\infty\)-smooth bump function. Thus, as the author points out, the Asplund space \(C_0(D)\) cannot be used to settle the long-standing open question of whether every Asplund space admits a \(C^1\)-smooth bump function.