Multiplicative maps of ordered Jordan algebras (Q1311062)

A Jordan \(*\)-map between \(C^*\)-algebras is defined as a one-to-one map that preserves involution and the Jordan product \(x\circ y={1\over 2} (xy+ yx)\). It was shown by Hakeda and Saito that if \({\mathfrak U}_ 1\) and \({\mathfrak U}_ 2\) are von Neumann algebras (or \(\text{AW}^*\)-algebras) and \({\mathfrak U}_ 1\) contains no central Abelian projections, then any Jordan \(*\)-map between \({\mathfrak U}_ 1\) and \({\mathfrak U}_ 2\) is additive. Hakeda asked whether an analog of that result could be established for JBW-algebras (Jordan-Banach algebras); this problem was solved by the first two authors in [Dokl. Akad. Nauk UzSSR 1990, No. 10, 5-6 (1990; Zbl 0723.46046)]. This result will now be generalized to a broader class -- ordered Jordan algebras (OJ-algebras); incidentally, our proof will be more algebraic than the previous ones.