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

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 $(\ast )$ has a boundary with property $(\ast )$, hence it is isomorphically polyhedral.