Complemented copies of ℓ1 in Banach spaces with an unconditional basis

Let X be a Banach space with an unconditional basis. If X contains an isomorphic copy Y of ℓ1, then it contains a complemented copy of ℓ1 located inside Y (Theorem 1). The proof is based on the possibility of constructing a projection onto a copy of ℓ1 in X, or in a Banach function space, when the ranges of the unit vectors of ℓ1 are pairwise disjoint (Lemma 1). The latter result applies also to Orlicz spaces. We also show that if U is a complemented copy of ℓ1 in a Banach space W and Y⊂W is a “slightly perturbated” copy of U, then Y is complemented in W (Lemma 2).