On embeddings in the intersection X ∧ L∞

Abstract

Let X be a separable rearrangement invariant space on (0,infinity). If the intersection (X boolean AND L-infinity)(0,infinity) contains a complemented subspace isomorphic to & ell,(2), then X contains a complemented sublattice lattice-isomorphic to & ell,(2). Moreover, we prove that the space (X + L-infinity)(0,infinity) cannot be isomorphically embedded into (X boolean AND L-infinity)(0,infinity) as a complemented subspace provided that X has nontrivial Rademacher cotype.