Szlenk and $w^*$-dentability indices of $C(K)$

Given any compact, Hausdorff space $K$ and $1<p<\infty$, we compute the Szlenk and $w^*$-dentability indices of the spaces $C(K)$ and $L_p(C(K))$. We show that if $K$ is compact, Hausdorff, scattered, $CB(K)$ is the Cantor-Bendixson index of $K$, and $ξ$ is the minimum ordinal such that $CB(K)\leqslant ω^ξ$, then $Sz(C(K))=ω^ξ$ and $Dz(C(K))=Sz(L_p(C(K)))= ω^{1+ξ}.$

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