A small Banach space $C(K)$ without nice renormings

Abstract

We prove that consistently $\ddot{I}{‰}_1<\mathfrak{c}$ and there exists a compact space $K$ whose Banach space $C(K)$ of continuous real-valued functions is Grothendieck, has density $\ddot{I}{‰}_1$, and admits no renorming which is strictly convex or sequentially Kadets--Klee.