Polyhedrality for twisted sums with $C(\omega^\alpha)$

TL;DR

This paper investigates conditions under which twisted sums of certain Banach spaces retain polyhedrality, providing partial solutions to the 3-space problem for isomorphic polyhedrality in specific classes of spaces.

Contribution

It offers new partial answers to the 3-space problem for isomorphic polyhedrality, focusing on twisted sums involving $C(eta)$ and spaces with property $(*)$.

Findings

Twisted sums of $C(eta)$ with separable isomorphically polyhedral spaces with BAP are isomorphically polyhedral.

Twisted sums of $c_0( ext{cardinality})$ with spaces having a boundary with property $(*)$ also have a boundary with property $(*)$.

Abstract

We obtain two partial answers to the 3-space problem for isomorphic polyhedrality: (1) every twisted sum of $C(\alpha)$, $\alpha<\omega_1$, with a separable isomorphically polyhedral space with the BAP, is isomorphically polyhedral. (2) Every twisted sum of $c_0(\kappa)$ and a Banach space having a boundary with property $(*)$ has a boundary with property $(*)$, hence it is isomorphically polyhedral.