Weak∗ dentability index of spaces C([0,α])

We compute the weak∗-dentability index of the spaces C(K) where K is a countable compact space. Namely Dz(C([0,ωωα]))=ω1+α+1, whenever 0⩽α<ω1. More generally, Dz(C(K))=ω1+α+1 if K is a scattered compact whose height η(K) satisfies ωα<η(K)⩽ωα+1 with an α countable.