On complemented copies of 𝑐₀ in 𝐿^{𝑝}_{𝑋},1≤𝑝<∞

Abstract

Let (5, E,/z) be a not purely atomic measure space and X be a Banach space. In this note we want to show that if X contains a copy of Co then the usual Banach space of the Lebesgue-Bochner integrable functions Lvx, 1 < p < oo, contains a complemented copy of Co-Our result is similar in spirit to one obtained in [1] by Cembranos concerning the Banach space Cx(if); in passing we observe that the Cembranos result has been extended in [3] to the case of e-tensor products and then in [2] to the case of the Banach space of compact weak*-weak continuous operators.In order to prove our theorem we need the definition of limited sets. A (bounded)