A note on the Banach–Mazur distances between c0 and other ℓ1 -preduals

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 \(X^*\).