On the axiomatic definition of real JB∗–triples

In the last twenty years, a theory of real Jordan triples has been developed. In 1994 T. Dang and B. Russo introduced the concept of J∗B–triple. These J∗B–triples include real C∗–algebras and complex JB∗–triples. However, concerning J∗B–triples, an important problem was left open. Indeed, the question was whether the complexification of a J∗B–triple is a complex JB∗–triple in some norm extending the original norm. T. Dang and B. Russo solved this problem for commutative J∗B–triples. In this paper we characterize those J∗B–triples with a unitary element whose complexifications are complex JB∗–triples in some norm extending the original one. We actually find a necessary and sufficient new axiom to characterize those J∗B–triples with a unitary element which are J∗B–algebras in the sense of [1] or real JB∗–triples in the sense of [4].