Diameter two properties and the Radon–Nikodým property in Orlicz spaces

Abstract

Some necessary and sufficient conditions are found for Banach function lattices to have the Radon–Nikodým property. Consequently it is shown that an Orlicz function space Lφ over a non-atomic σ-finite measure space (Ω,Σ,μ), not necessarily separable, has the Radon–Nikodým property if and only if φ is an N-function at infinity and satisfies the appropriate Δ2 condition. For an Orlicz sequence space ℓφ, it has the Radon–Nikodým property if and only if φ satisfies the Δ20 condition. In the second part a relationship between uniformly ℓ12 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 Lφ has the Daugavet property only if φ is linear, so when Lφ is isometric to L1. 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 φ.