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

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$0\le\alpha<\omega_1$\mathrm{.}Moregenerally,${Dz}(C(K))=\omega^{1+\alpha+1}$if$K$isascatteredcompactwhoseheight$\eta(K)$satisfies$\omega^\alpha<\eta(K)\leq \omega^{\alpha+1}$withan$\alpha$ countable.