ZFC construction of separable scattered compact spaces without smooth renormings

Construct, in ZFC, a compact, scattered, separable space K such that C(K) admits no equivalent Fréchet smooth norm; additionally determine whether K can be chosen so that C(K) has the renorming-stable ball-covering property, and whether the conclusion can be strengthened by replacing Fréchet smoothness with Gâteaux smoothness.

Background

The paper gives examples of compact separable scattered spaces K for which C(K) has the renorming-stable ball-covering property (RBCP) but lacks an equivalent norm with strictly convex dual. It also presents, consistently under a forcing model, a compact separable scattered space whose C(K) has the RBCP but admits no equivalent Fréchet smooth norm. The authors ask whether a corresponding example exists in ZFC, and whether the same phenomenon can be obtained for Gâteaux smooth renormings.

The problem is motivated by the contrast with Haydon’s ZFC example of a non-separable scattered compact space whose C(K) admits neither an equivalent strictly convex norm nor an equivalent Gâteaux smooth norm, together with the obstruction that separable compactifications of omega have w*-separable duals and therefore admit equivalent strictly convex norms. The RBCP requirement would connect the renorming obstruction to the central ball-covering property studied in the paper.