Geodesic
On \(\ell_1\)-preduals distant by 1
Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 72 (2018) no. 2.

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For every predual X of ℓ_1 such that the standard basis in ℓ_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 ℓ_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, σ(Y^*,Y)-FPP) whenever d(X,Y)=1. Then, we construct a predual X of ℓ_1 for which ℓ_1 lacks the stable σ(ℓ_1,X)-FPP but it has almost stable σ(ℓ_1,X)-FPP, which in turn is a strictly stronger property than the σ(ℓ_1,X)-FPP. Finally, in the general setting of preduals of ℓ_1, we give a sufficient condition for almost stable weak^* fixed point property in ℓ_1 and we prove that for a wide class of spaces this condition is also necessary.
Keywords: Banach-Mazur distance, nearly (almost) isometric Banach spaces, \(\ell_1\)-preduals, hyperplanes in c, weak\(^*\) fixed point property, stable weak\(^*\) fixed point property, almost stable weak\(^*\) fixed point property, nonexpansive mappings 
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Piasecki, Łukasz. On \(\ell_1\)-preduals distant by 1. Annales Universitatis Mariae Curie-Skłodowska. Mathematica , Tome 72 (2018) no. 2. http://geodesic.mathdoc.fr/item/AUM_2018_72_2_a2/

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