Explicit models of ℓ₁-preduals and the weak^* fixed point property in ℓ₁

2022/09/12 by Emanuele Casini, Casini, Emanuele, Enrico Miglierina +3
Computer Science · Mathematics · #46B25 #47H10 #Advanced Differential Equations and Dynamical Systems #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Optimization and Variational Analysis

paper · pdf · doi:10.48550/arxiv.2209.05116

openalex publication_date 2022/09/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

We provide a concrete isometric description of all the preduals of ℓ₁ for which the standard basis in ℓ₁ has a finite number of w^*-limit points. Then, we apply this result to give an example of an ℓ₁-predual X such that its dual X^* lacks the weak^* fixed point property for nonexpansive mappings (briefly, w^*-FPP), but X does not contain an isometric copy of any hyperplane W_α of the space c of convergent sequences such that W_α is a predual of ℓ₁ and W_α^* lacks the w^*-FPP. This answers a question left open in the 2017 paper of the present authors.

Related

• Subspaces of separable L₁-preduals: W_α everywhere
• Stability constants of the weak^* fixed point property for the space ℓ₁
• Hyperplanes in the space of convergent sequences and preduals of ℓ₁
• Separable Lindenstrauss spaces whose duals lack the weak^* fixed point property for nonexpansive mappings
• Embedding ℓ₂ and J into subspaces of JT and JT^*