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On real forms of JB\(^*\)-triples. (English) Zbl 0834.17047
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 T. Dang for complex \(\text{JB}^*\)-triples [Proc. Am. Math. Soc. 114, 971-980 (1992; Zbl 0773.46025)].

MSC:
17C65 Jordan structures on Banach spaces and algebras
46H70 Nonassociative topological algebras
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