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Szlenk and $w^*$-dentability indices of $C(K)$ (1605.01969v1)
Published 6 May 2016 in math.FA
Abstract: 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 $\xi$ is the minimum ordinal such that $CB(K)\leqslant \omega\xi$, then $Sz(C(K))=\omega\xi$ and $Dz(C(K))=Sz(L_p(C(K)))= \omega{1+\xi}.$