$M$-ideals, yet again: the case of real JB$^*$-triples
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914637321076736 |
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| author | Blecher, David P. Neal, Matthew Peralta, Antonio M. Su, Shanshan |
| author_facet | Blecher, David P. Neal, Matthew Peralta, Antonio M. Su, Shanshan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_05565 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $M$-ideals, yet again: the case of real JB$^*$-triples Blecher, David P. Neal, Matthew Peralta, Antonio M. Su, Shanshan Operator Algebras Primary 46B04, Secondary 46B20, 46L57, 47L05, 47C05, 17C65 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. |
| title | $M$-ideals, yet again: the case of real JB$^*$-triples |
| topic | Operator Algebras Primary 46B04, Secondary 46B20, 46L57, 47L05, 47C05, 17C65 |
| url | https://arxiv.org/abs/2401.05565 |