Weak${}^{\ast }$ dentability index of spaces $C([0,\alpha ])$

Abstract

We compute the weak${}^{\ast }$-dentability index of the spaces $C(K)$ where $K$ is a countable compact space. Namely $Dz(C([0,{\omega }^{{\omega }^{\alpha }}]))={\omega }^{1+\alpha +1}$, whenever $0\le \alpha <{\omega }_1$. More generally, $Dz(C(K))={\omega }^{1+\alpha +1}$ if $K$ is a scattered compact whose height $\eta (K)$ satisfies ${\omega }^{\alpha }<\eta (K)\le {\omega }^{\alpha +1}$ with an $\alpha$ countable.