Subspaces of separable $L_1$-preduals: $W_α$ everywhere
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913191316946944 |
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| author | Casini, Emanuele Miglierina, Enrico Piasecki, Łukasz |
| author_facet | Casini, Emanuele Miglierina, Enrico Piasecki, Łukasz |
| contents | The spaces $W_α$ are the Banach spaces whose duals are isometric to $\ell_1$ and such that the standard basis of $\ell_1$ is $w^*$-convergent to $α\in \ell_1$. The core result of our paper proves that an $\ell_1$-predual $X$ contains isometric copies of all $W_α$, where the norm of $α$ is controlled by the supremum of the norms of the $w^*$-cluster points of the extreme points of the closed unit ball in $\ell_1$. More precisely, for every $\ell_1$-predual $X$ we have
$$
r^*(X)=\sup\left\lbrace \left\|g^*\right\|: g^*\in \left(\mathrm{ext}\, B_{\ell_1}\right)'\right\rbrace =\sup \left\lbrace \left\| α\right\|: \, α\in B_{\ell_1}, \, W_α\subset X\right\rbrace . $$ We also prove that, for any $\varepsilon >0$, $X$ contains an isometric copy of some space $W_α$ with $\left\| α\right\|>r^*(X)- \varepsilon$ which is $(1+ \varepsilon)$-complemented in $X$. From these results we obtain several consequences. First we provide a new characterization of separable $L_1$-preduals containing an isometric copy of a space of affine continuous functions on a Choquet simplex. Then, we prove that an $\ell_1$-predual $X$ contains almost isometric copies of the space $c$ of convergent sequences if and only if $X^*$ lacks the stable $w^*$-fixed point property for nonexpansive mappings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04819 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Subspaces of separable $L_1$-preduals: $W_α$ everywhere Casini, Emanuele Miglierina, Enrico Piasecki, Łukasz Functional Analysis 46B04, 46B45, 47H10 The spaces $W_α$ are the Banach spaces whose duals are isometric to $\ell_1$ and such that the standard basis of $\ell_1$ is $w^*$-convergent to $α\in \ell_1$. The core result of our paper proves that an $\ell_1$-predual $X$ contains isometric copies of all $W_α$, where the norm of $α$ is controlled by the supremum of the norms of the $w^*$-cluster points of the extreme points of the closed unit ball in $\ell_1$. More precisely, for every $\ell_1$-predual $X$ we have $$ r^*(X)=\sup\left\lbrace \left\|g^*\right\|: g^*\in \left(\mathrm{ext}\, B_{\ell_1}\right)'\right\rbrace =\sup \left\lbrace \left\| α\right\|: \, α\in B_{\ell_1}, \, W_α\subset X\right\rbrace . $$ We also prove that, for any $\varepsilon >0$, $X$ contains an isometric copy of some space $W_α$ with $\left\| α\right\|>r^*(X)- \varepsilon$ which is $(1+ \varepsilon)$-complemented in $X$. From these results we obtain several consequences. First we provide a new characterization of separable $L_1$-preduals containing an isometric copy of a space of affine continuous functions on a Choquet simplex. Then, we prove that an $\ell_1$-predual $X$ contains almost isometric copies of the space $c$ of convergent sequences if and only if $X^*$ lacks the stable $w^*$-fixed point property for nonexpansive mappings. |
| title | Subspaces of separable $L_1$-preduals: $W_α$ everywhere |
| topic | Functional Analysis 46B04, 46B45, 47H10 |
| url | https://arxiv.org/abs/2401.04819 |