Diameter two properties and the Radon-Nikodym property in Orlicz spaces

Abstract

Some necessary and sufficient conditions are found for Banach function lattices to have the Radon- Nikodym property. Consequently it is shown that an Orlicz function space L-phi over a non-atomic sigma-finite measure space (Omega, Sigma, mu), not necessarily separable, has the Radon-Nikodym property if and only if phi is an N-function at infinity and satisfies the appropriate Delta(2) condition. For an Orlicz sequence space l(phi), it has the Radon-Nikodym property if and only if phi satisfies the Delta(0)(2) condition. In the second part a relationship between uniformly l(1)(2) points of the unit sphere of a Banach space and the diameter of the slices are studied. Using these results, a quick proof is given that an Orlicz space L-phi has the Daugavet property only if phi is linear, so when L-phi is isometric to L-1. Another consequence is that Orlicz spaces equipped with the Orlicz norm generated by N-functions never have the local diameter two property, while it is well-known that when equipped with the Luxemburg norm, it may have that property. Finally, it is shown that the local diameter two property, the diameter two property, and the strong diameter two property are equivalent in Orlicz function and sequence spaces with the Luxemburg norm under appropriate conditions on phi. (C) Published by Elsevier B.V. on behalf of Royal Dutch Mathematical Society (KWG).