This is an announcement for the paper "Weak$^*$ dentability index of spaces $C([0,\alpha])$" by Petr Hajek, Gilles Lancien and Antonin Prochazka. Abstract: We compute the weak$^*$-dentability index of the spaces $C(K)$ where $K$ is a countable compact space. Namely $\mbox{Dz}(C([0,\omega^{\omega^\alpha}])) = \omega^{1+\alpha+1}$, whenever $0\le\alpha<\omega_1$. More generally, $\mbox{Dz}(C(K))=\omega^{1+\alpha+1}$ if $K$ is a scattered compact whose height $\eta(K)$ satisfies $\omega^\alpha<\eta(K)\leq \omega^{\alpha+1}$ with an $\alpha$ countable. Archive classification: math.FA Mathematics Subject Classification: 46B20, 46B03, 46E15 The source file(s), delta4revised.tex: 28188 bytes, is(are) stored in gzipped form as 0901.0681.gz with size 9kb. The corresponding postcript file has gzipped size 78kb. Submitted from: protony@centrum.cz The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0901.0681 or http://arXiv.org/abs/0901.0681 or by email in unzipped form by transmitting an empty message with subject line uget 0901.0681 or in gzipped form by using subject line get 0901.0681 to: math@arXiv.org.