C(K, E) Contains a Complemented Copy of c 0

Abstract

A Banach space E is said to have the Dunford-Pettis property if for every pair of weakly null sequences (x,) c E and (x') c E' one has lim(x,, x') 0. Following Diestel [1] we shall say that a Banach space E is hereditarily

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

“…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)…”

Transitivity in spaces of vector-valued functions

“…The space M could be isomorphic to a C 0 (L), but not to any C 0 (L, X) unless L is finite or X is finite dimensional. And this is so because every operator from M to a separable Banach space is weakly compact (by the results in [2]), while C 0 (L, X) contains a complemented isomorph of c 0 as long as L is infinite and X is infinite dimensional: a well-known result by Cembranos [8].…”

“…And this is so because every operator from M to a separable Banach space is weakly compact (by the results in [2]), while C 0 (L, X) contains a complemented isomorph of c 0 as long as L is infinite and X is infinite dimensional: a well-known result by Cembranos [8]. I strongly suspect C(N * , M) is actually transitive but I have been unable to find a proof.…”

When is complemented in tensor products of ?

“…An immediate consequence of the classical Cembranos–Freniche theorem [, Main Theorem], [, Corollary 2.5] is the following result. Theorem For each we have .…”