Complemented copies of 1 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 1, then it contains a complemented copy of 1 located inside Y (Theorem 1). The proof is based on the possibility of constructing a projection onto a copy of 1 in X, or in a Banach function space, when the ranges of the unit vectors of 1 are pairwise disjoint (Lemma 1). The latter result applies also to Orlicz spaces.We also show that if U is a complemented copy of 1 in a Banach space W and Y ⊂W is a “slightly perturbated” copy of U, then Y is complemented in W (Lemma 2). © 2007 Published by Elsevier Inc