An order theoretic characterization of spin factors

The famous Koecher–Vinberg theorem characterizes the Euclidean Jordan algebras among the finite dimensional order unit spaces as the ones that have a symmetric cone. Recently, Walsh gave an alternative characterization of the Euclidean Jordan algebras. He showed that the Euclidean Jordan algebras correspond to the finite dimensional order unit spaces (V, C, u) for which there exists a bijective map \(g : C° → C°\) with the property that \(g\) is antihomogeneous, that is, \(g (\lambda x) = \lambda^{-1}g(x)\) for all \(\lambda > 0\) and \( x \in C°\), and \(g\) is an order-antimorphism, that is, \(x ≤ c\space y\) if and only if \(g(y) ≤ g (x)\). In this paper, we make a first step towards extending this order theoretic characterization to infinite dimensional JB-algebras. We show that if \((V, C, u)\) is a complete order unit space with a strictly convex cone and \(dim V ≥ 3\), then there exists a bijective antihomogeneous order-antimorphism \(g : C° → C°\) if and only if \((V, C, u)\) is a spin factor.