  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 $g\colon C^\circ\to C^\circ$ 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.,
$x\leq_C y$ if and only if $g(y)\leq_C g(x)$. 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 $\dim V\geq 3$, then there exists a
bijective antihomogeneous order-antimorphism $g\colon C^\circ\to C^\circ$ if
and only if $(V,C,u)$ is a spin factor.
