Boundaries, Martin's Axiom, and (P)-properties in dual Banach spaces

Antonio S. Granero, and Juan M. Hernández. "Boundaries, Martin's Axiom, and (P)-properties in dual Banach spaces." Commentationes Mathematicae 56.1 (2016): null. <http://eudml.org/doc/292437>.

@article{AntonioS2016,
abstract = {Let $X$ be a Banach space and $\mathcal \{S\} \mathit \{eq\}(X^\{**\})$ (resp., $X_\{\aleph _0\}$) the subset of elements $\psi \in X^\{**\}$ such that there exists a sequence $(x_n)_\{n\ge 1\}\subset X$ such that $x_n\rightarrow \psi $ in the $w^*$-topology of $X^\{**\}$ (resp., there exists a separable subspace $Y\subset X$ such that $\psi \in \smash\{\{\overline\{Y\}^\{w^*\}\}\}$). Then: (i) if $\operatorname\{Dens\}(X)\ge \aleph _1$, the property $X^\{**\}=X_\{\aleph _0\}$ (resp., $X^\{**\}=\mathcal \{S\}\mathit \{eq\}(X^\{**\})$) is $\aleph _1$-determined, i.e., $X$ has this property iff $Y$ has, for every subspace $Y\subset X$ with $\operatorname\{Dens\}(Y)=\aleph _1$; (ii) if $X^\{**\}=X _\{\aleph _0\}$, $ (B(X^\{**\}),w^*)$ has countable tightness; (iii) under the Martin’s axiom $\mathit \{MA\} (\omega _1)$ we have $X^\{**\}=\mathcal \{S\}\mathit \{eq\}(X^\{**\})$ iff $(B(X^*),w^*)$ has countable tightness and $\\overline \{\text\{co\}\}(B)=\overline\{\text\{co\}\} ^\{w^*\}(K)$ for every subspace $Y\subset X$, every $w^*$-compact subset $K$ of $Y^*$, and every boundary $B\subset K$.},
author = {Antonio S. Granero, Juan M. Hernández},
journal = {Commentationes Mathematicae},
keywords = {Boundaries, Martin’s Axiom, equality $Seq(X^\{**\})=X^\{**\}$, super-(P) property},
language = {eng},
number = {1},
pages = {null},
title = {Boundaries, Martin's Axiom, and (P)-properties in dual Banach spaces},
url = {http://eudml.org/doc/292437},
volume = {56},
year = {2016},
}

TY - JOUR
AU - Antonio S. Granero
AU - Juan M. Hernández
TI - Boundaries, Martin's Axiom, and (P)-properties in dual Banach spaces
JO - Commentationes Mathematicae
PY - 2016
VL - 56
IS - 1
SP - null
AB - Let $X$ be a Banach space and $\mathcal {S} \mathit {eq}(X^{**})$ (resp., $X_{\aleph _0}$) the subset of elements $\psi \in X^{**}$ such that there exists a sequence $(x_n)_{n\ge 1}\subset X$ such that $x_n\rightarrow \psi $ in the $w^*$-topology of $X^{**}$ (resp., there exists a separable subspace $Y\subset X$ such that $\psi \in \smash{{\overline{Y}^{w^*}}}$). Then: (i) if $\operatorname{Dens}(X)\ge \aleph _1$, the property $X^{**}=X_{\aleph _0}$ (resp., $X^{**}=\mathcal {S}\mathit {eq}(X^{**})$) is $\aleph _1$-determined, i.e., $X$ has this property iff $Y$ has, for every subspace $Y\subset X$ with $\operatorname{Dens}(Y)=\aleph _1$; (ii) if $X^{**}=X _{\aleph _0}$, $ (B(X^{**}),w^*)$ has countable tightness; (iii) under the Martin’s axiom $\mathit {MA} (\omega _1)$ we have $X^{**}=\mathcal {S}\mathit {eq}(X^{**})$ iff $(B(X^*),w^*)$ has countable tightness and $\\overline {\text{co}}(B)=\overline{\text{co}} ^{w^*}(K)$ for every subspace $Y\subset X$, every $w^*$-compact subset $K$ of $Y^*$, and every boundary $B\subset K$.
LA - eng
KW - Boundaries, Martin’s Axiom, equality $Seq(X^{**})=X^{**}$, super-(P) property
UR - http://eudml.org/doc/292437
ER -