The Giesy--James theorem for general index \(p\), with an application to operator ideals on the \(p\)th James space

Abstract

A theorem of Giesy and James states that \(c_0\) is finitely representable in James' quasi-reflexive Banach space \(J_2\). We extend this theorem to the \(p\)th quasi-reflexive James space \(J_p\) for each \(p \in (1,\infty)\). As an application, we obtain a new closed ideal of operators on \(J_p\), namely the closure of the set of operators that factor through the complemented subspace \((\ell_\infty^1 \oplus \ell_\infty^2 \oplus...\oplus \ell_\infty^n \oplus...)_{\ell_p}\) of \(J_p\).