On embeddings in the intersection \(X\cap L_{\infty }\)

Let X be a separable rearrangement invariant space on \((0,\infty )\) $(0,\infty )$ . If the intersection \((X \cap L_{\infty })(0,\infty )\) $(X\cap L_{\infty })(0,\infty )$ contains a complemented subspace isomorphic to \({\ell }_2\) ${\ell }_2$ , then X contains a complemented sublattice lattice-isomorphic to \({\ell }_2\) ${\ell }_2$ . Moreover, we prove that the space \((X+L_{\infty })(0,\infty )\) $(X+L_{\infty })(0,\infty )$ cannot be isomorphically embedded into \((X \cap L_{\infty })(0,\infty )\) $(X\cap L_{\infty })(0,\infty )$ as a complemented subspace provided that X has nontrivial Rademacher cotype.