Subspaces generated by translations in rearrangement invariant spaces

In the setting of rearrangement invariant (r.i.) Banach function spaces E o n [0, infinity) we study the complementability of subspaces 0, generated by sequences of translations of Functions a is an element of E[0, 1). An r.i. Function space E is said to be nice e tin short, E is an element of N) if every subspace of type Q(a) is complemented. We give necessary and sufficie nt conditions for an r.i. function space to be nice. We determinate the Orl icz, Lorentz and Marcinkiewicz spaces belonging to the class.i'. As an appl ication we obtain a new characterization of the L-p-spilces, 1 < p < infini ty among the class of r.i. function spaces. (C) 1999 Academic Press.