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

Ryan M Causey

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

Functional Analysis 2016-05-09 v1

Authors: Ryan M Causey

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

Keywords

Szlenk index w*-dentability index C(K) space L_p space scattered compact space Cantor-Bendixson index ordinal

Cite

@article{arxiv.1605.01969,
  title  = {Szlenk and $w^*$-dentability indices of $C(K)$},
  author = {Ryan M Causey},
  journal= {arXiv preprint arXiv:1605.01969},
  year   = {2016}
}

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