On the Zelmanovian classification of prime \(\text{JB}^*\)-triples
Complex \(\text{JB}^*\)-triples are complex Banach spaces endowed with a triple product subjected to suitable conditions of algebraic and analytic nature. They were introduced by W. Kaup in order to provide an algebraic setting for the study of bounded symmetric domains in complex Banach spaces: bounded symmetric domains are essentially the open unit balls of complex \(\text{JB}^*\)-triples. Applying techniques developed by E. Zelmanov and later refined by A. D'Amour and K. McCrimmon, the authors prove a classification theorem for prime complex \(\text{JB}^*\)-triples, which roughly speaking asserts that prime complex \(\text{JB}^*\)-triples which are neither spin factors nor exceptional Cartan factors, are essentially either prime ternary rings of operators or Hermitian parts of prime ternary rings of operators. Real \(\text{JB}^*\)-triples are defined as norm-closed real subtriples of complex \(\text{JB}^*\)-triples. They were introduced in a paper by J. M. Isidro, W. Kaup and A. Rodríguez, where as main result it is proved that bijective linear mappings between real \(\text{JB}^*\)-triples are isometric if and only if they preserve the cube mapping. The authors also apply Zelmanovian techniques to obtain the corresponding classification theorem for prime real \(\text{JB}^*\)-triples.
scientific article; zbMATH DE number 1441824
scientific article