A note on the Banach–Mazur distances between \(c_0\) and other \(\ell_1\)-preduals

Agnieszka Gergont

Abstract


We prove that if \(X\) is an \(\ell_{1}\)-predual isomorphic to the space \(c_{0}\) of sequences converging to zero, then for any isomorphism \(T:X\rightarrow c_{0}\) we have \(\Vert T\Vert\, \Vert T^{-1}\Vert\ge1+2r^{*}(X)\), where \(r^{*}(X)\) is the smallest radius of the closed ball of the dual space \(X^{*}\) containing  all the weak\(^{*}\) cluster points of the set of all extreme points of the closed unit ball of  \(X^*\).

Keywords


\(\ell_1\)-preduals; Banach--Mazur distance; \(c_0\) space

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References


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DOI: http://dx.doi.org/10.17951/a.2022.76.1.25-30
Date of publication: 2022-10-05 20:39:32
Date of submission: 2022-10-04 19:08:19


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