Twisted sums of c ₀(I)

Abstract

We study in this paper a few remarkable properties of twisted sums Z(κ, X) of c ₀(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 c ₀(κ) and c ₀(I) are either subspaces of ℓ _∞(κ) or contain a complemented copy of c ₀(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 c ₀ 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.