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arXiv · 2401.05565

$M$-ideals, yet again: the case of real JB$^*$-triples

David P. Blecher·Matthew Neal·Antonio M. Peralta·Shanshan Su
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Abstract

We prove that a subspace of a real JBW$^*$-triple is an $M$-summand if and only if it is a weak$^*$-closed triple ideal. As a consequence, $M$-ideals of real JB$^*$-triples correspond to norm-closed triple ideals. As in the setting of complex JB$^*$-triples, a geometric property is characterized in purely algebraic terms. This is a newfangled treatment of the classical notion of $M$-ideal in the real setting by a fully new approach due to the unfeasibility of the known arguments in the setting of complex C$^*$-algebras and JB$^*$-triples. The results in this note also provide a full characterization of all $M$-ideals in real C$^*$-algebras, real JB$^*$-algebras and real TROs.

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David P. Blecher, Matthew Neal, Antonio M. Peralta, Shanshan Su. 2024-01-10. $M$-ideals, yet again: the case of real JB$^*$-triples. https://arxiv.org/abs/2401.05565

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