Rademacher series and isomorphisms of rearrangement invariant spaces on the finite interval and on the semi-axis

Let X be a rearrangement invariant function space on [0, 1].We consider the subspace RadiX of X which consists of all functions of the form f = ∞ k=1 xkrk, where xk are arbitrary independent functions from X and rk are usual Rademacher functions independent of {xk }. We prove that RadiX is complemented in X if and only if both X and its Köthe dual space X possess the so-called Kruglov property. As a consequence we show that the last conditions guarantee that X is isomorphic to some rearrangement invariant function space on [0,∞). This strengthens earlier results derived in different approach in [W.B. Johnson, B. Maurey, G. Schechtman, L. Tzafriri, Symmetric structures in Banach spaces, Mem. Amer. Math. Soc. 1 (217) (1979)]. © 2010 Elsevier Inc. All rights reserved.