Fixed points of reflections of compact convex sets and a   characterization of state spaces of Jordan Banach algebras
  In the present article we prove a fixed point theorem for reflections of compact convex sets and give a new characterization of state space of JB-algebras among compact convex sets. Namely they are exactly those compact convex sets which are strongly spectral and symmetric. 
Sh.A.Ayupov and N.J.Yadgorov
math.FA