On isomorphic embeddings of c into L1-preduals and some applications

Abstract

First we study the distortion ‖T‖‖T−1‖ of an isomorphic embedding T of the space c of convergent sequences into an infinite-dimensional L1-predual X such that (extBX⁎)′⊂rBX⁎ for some r∈(0,1), that is, the set of all weak⁎ cluster points of the set of all extreme points of the closed unit ball of the dual space X⁎ is contained in the closed ball of radius r∈(0,1). We prove that ‖T‖‖T−1‖≥(3−r)/(1+r). Moreover, we give some examples showing that for every r∈(0,1) our bound is optimal. Then, we apply our theorem to provide stability results for polyhedrality and extendability of compact operators in the setting of L1-preduals. To achieve the latter goal, we additionally give a new characterization of infinite-dimensional Banach spaces having the compact norm-preserving extension property for compact operators. Furthermore, in the framework of ℓ1-preduals such that the standard basis in ℓ1 is weak⁎ convergent, we provide precise values of stability constants for the weak⁎ fixed point property.