On the Zelmanovian classification of prime \(\text{JB}^*\)-triples
type: Item
instance of: scholarly article
title: On the Zelmanovian classification of prime \(\text{JB}^*\)-triples
zbMATH Open document ID: 0979.46050
author: A. Moreno
author: Ángel Rodríguez Palacios
published in: Journal of Algebra
publication date: 2000-05-08T00:00:00Z
review text: 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.
reviewed by: Antonio Fernández López
Mathematics Subject Classification ID: 46L70
Mathematics Subject Classification ID: 17C65
zbMATH DE Number: 1441824
zbMATH Keywords: \(\text{JB}^*\)-triple
zbMATH Keywords: prime ternary rings
zbMATH Keywords: bounded symmetric domains
zbMATH Keywords: Zelmanovian techniques
MaRDI profile type: Publication
full work available at URL: jabr.1999.8211
OpenAlex ID: W2016229157
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DOI: 10.1006/JABR.1999.8211
description: scientific article; zbMATH DE number 1441824