Weak$^*$ dentability index of spaces $C([0,\alpha])$

Petr Hajek (1) , Gilles Lancien (2) , Antonin . Prochazka (3)

1 IM / CAS - Institute of Mathematics of the Czech Academy of Science
2 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
3 IMB - Institut de Mathématiques de Bordeaux

We compute the weak$^*$-dentability index of the spaces $C(K)$ where $K$ is a countable compact space. Namely $\mbox{Dz}(C([0,\omega^{\omega^\alpha}])) = \omega^{1+\alpha+1}$, whenever $0\le\alpha<\omega_1$. More generally, $\mbox{Dz}(C(K))=\omega^{1+\alpha+1}$ if $K$ is a scattered compact whose height $\eta(K)$ satisfies $\omega^\alpha<\eta(K)\leq \omega^{\alpha+1}$ with an $\alpha$ countable.

Mots clés: Szlenk index; dentability index

Domaines: Analyse fonctionnelle [math.FA]

Petr Hajek, Gilles Lancien, Antonin . Prochazka. Weak$^*$ dentability index of spaces $C([0,\alpha])$. Journal of Mathematical Analysis and Applications, 2009, 353, pp.239-243. ⟨10.1016/j.jmaa.2008.11.075⟩. ⟨hal-00350187⟩