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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 completely new approach necessitated by the unfeasibility of the known arguments from 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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This content will become publicly available on March 4, 2026
M$M$‐Ideals in real operator algebras
Abstract In a recent paper, we showed that a subspace of a real ‐triple is an ‐summand if and only if it is a ‐closed triple ideal. As a consequence, ‐ideals of real ‐triples, including real ‐algebras, real ‐algebras and real TROs, correspond to norm‐closed triple ideals. In this paper, we extend this result by identifying the ‐ideals in (possibly non‐self‐adjoint) real operator algebras and Jordan operator algebras. The argument for this is necessarily different. We also give simple characterizations of one‐sided ‐ideals in real operator algebras, and give some applications to that theory.
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- Award ID(s):
- 2154903
- PAR ID:
- 10576354
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Mathematische Nachrichten
- ISSN:
- 0025-584X
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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