On real forms of JB\(^*\)-triples (Q1895710)

Real \(\text{JB}^*\)-triples are introduced as closed real subtriples of (complex) \(\text{JB}^*\)-triples. These may be realized as real forms of (complex) \(\text{JB}^*\)-triples. Real forms of (complex) \(\text{JBW}^*\)-triples are called by the authors real \(\text{JBW}^*\)- triples. Interesting geometrical properties linked with tripotents of real \(\text{JB}^*\)-triples and \(\text{JBW}^*\)-triples are given. The main theorem asserts that a bijective linear map \(G\) between real \(\text{JB}^*\)-triples is an isometry if and only if \(G(x^3)= G(x)^3\) for all \(x\). This generalizes a result of \textit{T. Dang} for complex \(\text{JB}^*\)-triples [Proc. Am. Math. Soc. 114, 971-980 (1992; Zbl 0773.46025)].