Order isomorphisms between cones of JB-algebras

In this paper we completely describe the order isomorphisms between cones of atomic JBW-algebras. Moreover, we can write an atomic JBW-algebra as an algebraic direct summand of the so-called engaged and disengaged part. On the cone of the engaged part every order isomorphism is linear and the disengaged part consists only of copies of $\mathbb{R}$. Furthermore, in the setting of general JB-algebras we prove the following. If either algebra does not contain an ideal of codimension one, then every order isomorphism between their cones is linear if and only if it extends to a homeomorphism, between the cones of the atomic part of their biduals, for a suitable weak topology.