1-COMPLEMENTED SUBSPACES OF SPACES WITH 1-UNCONDITIONAL BASES

BEATA RANDRIANANTOANINA

Abstract. We prove that if X is a complex strictly monotone sequence space with 1-unconditional basis, Y ⊆ X has no bands isometric to ℓ_{2}^{2} and Y is the range of norm-one projection from X, then Y is a closed linear span a family of mutually disjoint vectors in X.

We completely characterize 1-complemented subspaces and norm-one projections in complex spaces ℓ_{p}(ℓ_{q}) for 1 ≤ p, q < ∞.

Finally we give a full description of the subspaces that are spanned by a family of disjointly supported vectors and which are 1-complemented in (real or complex) Orlicz or Lorentz sequence spaces. In particular if an Orlicz or Lorentz space X is not isomorphic to ℓ_{p} for some 1 ≤ p < ∞ then the only subspaces of X which are 1-complemented and disjointly supported are the closed linear spans of block bases with constant coefficients.