On the Kunen–Shelah properties in Banach spaces

Tom 157 / 2003

Antonio S. Granero, Mar Jiménez, Alejandro Montesinos, José P. Moreno, Anatolij Plichko Studia Mathematica 157 (2003), 97-120 MSC: 46B20, 46B26. DOI: 10.4064/sm157-2-1

Streszczenie

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]).

Autorzy

  • Antonio S. GraneroDepartamento Análisis Matemático
    Facultad de Matemáticas
    Universidad Complutense
    Madrid 28040, Spain
    e-mail
  • Mar JiménezDepartamento Análisis Matemático
    Facultad de Matemáticas
    Universidad Complutense
    Madrid 28040, Spain
    e-mail
  • Alejandro MontesinosDepartamento Análisis Matemático
    Facultad de Matemáticas
    Universidad Complutense
    Madrid 28040, Spain
    e-mail
  • José P. MorenoDepartamento de Matemáticas
    Facultad de Ciencias
    Universidad Autónoma de Madrid
    Madrid 28049, Spain
    e-mail
  • Anatolij PlichkoDepartment of Mathematics
    Pedagogical University
    Shevchenko st. 1
    Kirovograd 2506, Ukraine
    e-mail

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