2024/01/09 by Emanuele Casini, Casini, Emanuele, Enrico Miglierina +3
Computer Science · Mathematics · #46B04 #46B45 #47H10 #Advanced Banach Space Theory #FOS: Mathematics #Fixed Point Theorems Analysis #Functional Analysis (math.FA) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2401.04819
openalex publication_date 2024/01/09 · openalex created_date 2024/01/13 · openalex updated_date 2026/07/28
The spaces Wα are the Banach spaces whose duals are isometric to ℓ1 and such that the standard basis of ℓ1 is w^*-convergent to α∈ ℓ1. The core result of our paper proves that an ℓ1-predual X contains isometric copies of all Wα, where the norm of α is controlled by the supremum of the norms of the w^*-cluster points of the extreme points of the closed unit ball in ℓ1. More precisely, for every ℓ1-predual X we have r^*(X)=sup\lbrace ‖g^*‖: g^*∈ (ext Bℓ1)'\rbrace =sup \lbrace ‖ α‖: α∈ Bℓ1, Wα⊂ X\rbrace . We also prove that, for any ε >0, X contains an isometric copy of some space Wα with ‖ α‖>r^*(X)- ε which is (1+ ε)-complemented in X. From these results we obtain several consequences. First we provide a new characterization of separable L1-preduals containing an isometric copy of a space of affine continuous functions on a Choquet simplex. Then, we prove that an ℓ1-predual X contains almost isometric copies of the space c of convergent sequences if and only if X^* lacks the stable w^*-fixed point property for nonexpansive mappings.