On l1-preduals distant by 1
Journal Title: Annales Universitatis Mariae Curie-Skłodowska. Sectio A, Mathematica - Year 2018, Vol 72, Issue 2
Abstract
For every predual X of l1 such that the standard basis in l1 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 l1, 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, s(Y *, Y )-FPP) whenever d(X; Y ) = 1. Then, we construct a predual X of l1 for which l1 lacks the stable s(l1,X)-FPP but it has almost stable s(l1;X)-FPP, which in turn is a strictly stronger property than the s(l1,X)-FPP. Finally, in the general setting of preduals of l1, we give a sufficient condition for almost stable weak* fixed point property in l1 and we prove that for a wide class of spaces this condition is also necessary.
Authors and Affiliations
Łukasz Piasecki
Keywords
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- EP ID EP530845
- DOI 10.17951/a.2018.72.2.41
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