An order theoretic characterization of spin factors

Abstract

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 $g:C^{\circ }\to C^{\circ }$ with the property that $g$ is antihomogeneous, i.e., $g(\lambda x)={\lambda }^{-1}g(x)$ for all $\lambda >0$ and $x\in C^{\circ }$, and $g$ is an order-antimorphism, i.e., $x{\le }_{C}y$ if and only if $g(y){\le }_{C}g(x)$. 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 $\dim V\ge 3$, then there exists a bijective antihomogeneous order-antimorphism $g:C^{\circ }\to C^{\circ }$ if and only if $(V,C,u)$ is a spin factor.

15 pages (minor revisions). To appear in Quarterly Journal of Mathematics