Weak∗ dentability index of spaces

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