Twisted sums of \(c_0(I)\)

The paper studies properties of twisted sums of a Banach space \(X\) with \(c_0(\kappa)\). We first prove a representation theorem for such twisted sums from which we will obtain, among others, the following: (a) twisted sums of \(c_0(I)\) and \(c_0(\kappa)\) are either subspaces of \(\ell_\infty(\k...

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Bibliographic Details
Published in:arXiv.org 2022-04
Main Authors: Castillo, Jesús M F, Alberto Salguero Alarcón
Format: Article
Language:English
Subjects:
Banach spaces
Continuity (mathematics)
Subspaces
Sums
Online Access:Get full text
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Summary:The paper studies properties of twisted sums of a Banach space \(X\) with \(c_0(\kappa)\). We first prove a representation theorem for such twisted sums from which we will obtain, among others, the following: (a) twisted sums of \(c_0(I)\) and \(c_0(\kappa)\) are either subspaces of \(\ell_\infty(\kappa)\) or trivial on a copy of \(c_0(\kappa^+)\); (b) under the hypothesis \([\mathfrak p = \mathfrak c]\), 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 \(C(K)\) with \(c_0(\kappa)\) that is not isomorphic to a space of continuous functions; (c) all such twisted sums are Lindenstrauss spaces when \(X\) is a Lindenstrauss space and \(G\)-spaces when \(X=C(K)\) with \(K\) convex, which shows tat a result of Benyamini is optimal; (d) they are isomorphically polyhedral when \(X\) is a polyhedral space with property (\(\star\)), which solves a problem of Castillo and Papini.
ISSN:2331-8422