An isomorphic characterization of L1-spaces

We show that a sequentially (τ)-complete topological vector lattice Xτ is isomorphic to some L1(μ), if and only if the positive cone can be written as X+ = ℝ+B for some convex, (τ)-bounded, and (τ)-closed set B ⊂ X+ \ {0}. The same result holds under weaker hypotheses, namely the Riesz decomposition property for X (not assumed to be a vector lattice) and the monotonic σ-completeness (monotonic Cauchy sequences converge). The isometric part of the main result implies the well-known representation theorem of Kakutani for (AL)-spaces. As an application we show that on a normed space Y of infinite dimension, the “ball-generated” ordering induced by the cone Y+ = ℝ+ (for ‖u‖ >) cannot have the Riesz decomposition property. A second application deals with a pointwise ordering on a space of multivariate polynomials.