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 in order to give an axiomatic definition
of JB*-triples over the real field. 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 Alverman
or real JB*-triples in the sense of those introduced by Isidro,
Kaup and Rodríguez.
Antonio M. Peralta < aperalta@ugr.es >