On the Kunen–Shelah properties in Banach spaces

Antonio S. Granero ¹   ; Mar Jiménez ¹   ; Alejandro Montesinos ¹   ; José P. Moreno ²   ; Anatolij Plichko ³

¹ Departamento Análisis Matemático Facultad de Matemáticas Universidad Complutense Madrid 28040, Spain
² Departamento de Matemáticas Facultad de Ciencias Universidad Autónoma de Madrid Madrid 28049, Spain
³ Department of Mathematics Pedagogical University Shevchenko st. 1 Kirovograd 2506, Ukraine

Studia Mathematica, Tome 157 (2003) no. 2, pp. 97-120

Résumé

We introduce and study the Kunen–Shelah properties ${\rm KS}_i$, $ i=0,1,\ldots ,7$. Let us highlight some of our results for a Banach space $X$: (1) $X^*$ has a $w^*$-nonseparable equivalent dual ball iff $X$ has an $\omega$-polyhedron (i.e., a bounded family $\{x_i\}_{i \omega}$ such that $x_j\not\in \overline{\rm co} (\{x_i :i\in \omega\setminus \{j\}\})$ for every $j\in \omega$) iff $X$ has an uncountable bounded almost biorthogonal system (UBABS) of type $\eta$ for some $\eta\in [0,1)$ (i.e., a bounded family $\{(x_{\alpha},f_{\alpha})\}_{1\leq \alpha \omega}\subset X\times X^*$ such that $f_{\alpha}(x_{\alpha})=1$ and $|f_{\alpha}(x_{\beta})|\leq\eta$ if $\alpha\neq\beta$); (2) if $X$ has an uncountable $\omega$-independent system then $X$ has an UBABS of type $\eta$ for every $\eta \in (0,1)$; (3) if $X$ does not have the property (C) of Corson, then $X$ has an $\omega$-polyhedron; (4) $X$ has no $\omega$-polyhedron iff $X$ has no convex right-separated $\omega$-family (i.e., a bounded family $\{x_i\}_{i \omega}$ such that $x_j\not\in \overline{\rm co} (\{x_i: j i \omega\})$ for every $j\in \omega$) iff every $w^*$-closed convex subset of $X^*$ is $w^*$-separable iff every convex subset of $X^*$ is $w^*$-separable iff $\mu (X)=1$, $\mu (X)$ being the Finet–Godefroy index of $X$ (see [1]).

Détail

DOI : 10.4064/sm157-2-1

Keywords: introduce study kunen shelah properties ldots highlight results banach space nbsp * has * nonseparable equivalent dual ball has omega polyhedron bounded family omega overline omega setminus every omega has uncountable bounded almost biorthogonal system ubabs type eta eta bounded family alpha alpha leq alpha omega subset times * alpha alpha alpha beta leq eta alpha neq beta nbsp nbsp has uncountable omega independent system has ubabs type eta every eta nbsp does have property corson has omega polyhedron nbsp has omega polyhedron has convex right separated omega family bounded family omega overline omega every omega every * closed convex subset * * separable every convex subset * * separable being finet godefroy index see nbsp