Twisted Sums of C0(I) 1

Abstract

We study in this paper a few remarkable properties of twisted sums Z(κ, X) of c0(κ) and a Banach space X. We first prove a representation  theorem for such twisted sums from which we will obtain, among others, the following: (a) twisted sums of c0(κ) and c0(I) are either  subspaces of ℓ∞(κ) or contain a complemented copy of c0(κ +); (b) under the hypothesis [p = c], when K is either a suitable Corson  compact, a separable Rosenthal compact or a scattered compact of finite height, there is a twisted sum of c0 and C(K) that is not  isomorphic to a space of continuous functions; (c) all twisted sums Z(κ, X) are isomorphically Lindenstrauss spaces when X is a  Lindenstrauss space; (d) all twisted sums Z(κ, X) are isomorphically polyhedral when X is a polyhedral space with a σ-discrete boundary,  which solves a problem of Castillo and Papini.