On the structure of non-dentable subsets of $C(\omega ^{\omega ^{{k}}} )$

Pericles D. Pavlakos¹ ; Minos Petrakis¹

¹Department of Sciences, Section of Mathematics Technical University of Crete 73100 Chania, Greece

Studia Mathematica, Tome 203 (2011) no. 3, pp. 205-222

Résumé

It is shown that there is no closed convex bounded non-dentable subset $K$ of $C(\omega ^{\omega ^{k}})$ such that on subsets of $K$ the PCP and the RNP are equivalent properties. Then applying the Schachermayer–Rosenthal theorem, we conclude that every non-dentable $K$ contains a non-dentable subset $L$ so that on $L$ the weak topology coincides with the norm topology. It follows from known results that the RNP and the KMP are equivalent on subsets of $C(\omega ^{\omega ^{k}})$.

Détail

DOI : 10.4064/sm203-3-1

Keywords: shown there closed convex bounded non dentable subset omega omega subsets pcp rnp equivalent properties applying schachermayer rosenthal theorem conclude every non dentable contains non dentable subset weak topology coincides norm topology follows known results rnp kmp equivalent subsets omega omega