Embeddability of L1(μ) in dual spaces, geometry of cones and a characterization of c0

In this article we suppose that (Ω,Σ,μ) is a measure space and T an one-to-one, linear, continuous operator of L1(μ) into the dual E of a Banach space E. For any measurable set A consider the image T (L+1 (μA)) of the positive cone of the space L1(μA) in E , where μA is the restriction of the measure μ on A. We provide geometrical conditions on the cones T (L+1 (μA)) which yield that the measure μ is atomic, i.e., that L1(μ) is lattice isometric to 1(A), where A denotes the set of atoms of μ. This result yields also a new characterization of c0(Γ ).  2003 Elsevier Inc. All rights reserved.