Polyhedrality for twisted sums with $C(ω^α)$

TL;DR

The paper investigates whether isomorphic polyhedrality is preserved in 3-space, focusing on twisted sums. It develops a general 3-space framework: if $X$ and a family of $Y_N$ are separable isomorphically polyhedral with $X$ having the BAP, and a subspace $Y$ of $c_0(\mathbb N, Y_N)$ yields polyhedral twisted sums, then any twisted sum with $Y$ does as well; this is then specialized to $Y$ built from $C(\alpha)$ spaces, giving isomorphically polyhedral twisted sums for all $\alpha<\omega_1$ under separability and BAP assumptions. The second main contribution corrects earlier claims about boundaries, clarifying the role of boundaries with property $(*)$ as the key sufficient condition for polyhedral renormings, and proving that twisted sums with $c_0(\kappa)$ preserve a boundary with $(*)$, hence are isomorphically polyhedral. Together, these results advance the understanding of the 3-space problem for polyhedrality by combining quasilinear map techniques, pullback/pushout arguments, and boundary preservation under twisted sums.

Abstract

We obtain two partial answers to the 3-space problem for isomorphic polyhedrality: (1) every twisted sum of $C(α)$, $α<ω_1$, with a separable isomorphically polyhedral space with the BAP, is isomorphically polyhedral. (2) Every twisted sum of $c_0(κ)$ and a Banach space having a boundary with property $(*)$ has a boundary with property $(*)$, hence it is isomorphically polyhedral.

Theorems & Definitions (13)

• Proposition 2.1
• proof
• Claim
• proof : Proof of the Claim
• Theorem 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof