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## Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica

2018 | 72 | 2 |
Tytuł artykułu

### On $$\ell_1$$-preduals distant by 1

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EN
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EN
For every predual $$X$$ of $$\ell_1$$ such that the standard basis in $$\ell_1$$ is weak$$^*$$ convergent, we give explicit models of all Banach spaces $$Y$$ for which the Banach-Mazur distance $$d(X,Y)=1$$. As a by-product of our considerations, we obtain some new results in metric fixed point theory. First, we show that the space $$\ell_1$$, with a predual $$X$$ as above, has the stable weak$$^*$$ fixed point property if and only if it has almost stable weak$$^*$$ fixed point property, i.e. the dual $$Y^*$$ of every Banach space $$Y$$ has the weak$$^*$$ fixed point property (briefly, $$\sigma(Y^*,Y)$$-FPP) whenever $$d(X,Y)=1$$. Then, we construct a predual $$X$$ of $$\ell_1$$ for which $$\ell_1$$ lacks the stable $$\sigma(\ell_1,X)$$-FPP but it has almost stable $$\sigma(\ell_1,X)$$-FPP, which in turn is a strictly stronger property than the $$\sigma(\ell_1,X)$$-FPP. Finally, in the general setting of preduals of $$\ell_1$$, we give a sufficient condition for almost stable weak$$^*$$ fixed point property in $$\ell_1$$ and we prove that for a wide class of spaces this condition is also necessary.
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EN
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wydano
2018
online
2018-12-22
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Bibliografia
• Alspach, D. E., A $$\ell_1$$-predual which is not isometric to a quotient of $$C(\alpha)$$, arXiv:math/9204215v1 [math.FA] 27 Apr. 1992.
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• Casini, E., Miglierina, E., Piasecki, Ł., Popescu, R., Stability constants of the weak$$^*$$ fixed point property in the space $$\ell_1$$, J. Math. Anal. Appl. 452 (1) (2017), 673-684.
• Casini, E., Miglierina, E., Piasecki, Ł., Popescu, R., Weak$$^*$$ fixed point property in $$\ell_1$$ and polyhedrality in Lindenstrauss spaces, Studia Math. 241 (2) (2018), 159-172.
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• Piasecki, Ł., On Banach space properties that are not invariant under the Banach-Mazur distance 1, J. Math. Anal. Appl. 467 (2018), 1129-1147.
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