On \(\ell_1\)-preduals distant by 1

Łukasz Piasecki

Abstract


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.

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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References


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DOI: http://dx.doi.org/10.17951/a.2018.72.2.41
Date of publication: 2018-12-22 22:03:13
Date of submission: 2018-12-21 22:49:21


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