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

TL;DR

This paper computes the Szlenk and $w^*$-dentability indices for spaces of continuous functions and their $L_p$-spaces over compact, scattered spaces, revealing their precise ordinal values based on the Cantor-Bendixson index.

Contribution

It provides explicit formulas for Szlenk and $w^*$-dentability indices of $C(K)$ and $L_p(C(K))$ spaces in terms of the Cantor-Bendixson index, extending understanding of their geometric properties.

Findings

Szlenk index of $C(K)$ equals $ ext{omega}^\xi$.

$w^*$-dentability index of $C(K)$ equals $ ext{omega}^{1+\xi}$.

Indices depend on the Cantor-Bendixson index of $K$.

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 $Dz(C(K))=Sz(L_p(C(K)))= \omega^{1+\xi}.$