M$M$‐Ideals in real operator algebras

David P. Blecher, Matthew Neal, Antonio M. Peralta and Shanshan Su

Mathematische Nachrichten, 2025, vol. 298, issue 4, 1328-1341

Abstract: In a recent paper, we showed that a subspace of a real JBW∗${\rm JBW}^*$‐triple is an M$M$‐summand if and only if it is a weak∗${\rm weak}^*$‐closed triple ideal. As a consequence, M$M$‐ideals of real JB∗${\rm JB}^*$‐triples, including real C∗${\rm C}^*$‐algebras, real JB∗${\rm JB}^*$‐algebras and real TROs, correspond to norm‐closed triple ideals. In this paper, we extend this result by identifying the M$M$‐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 M$M$‐ideals in real operator algebras, and give some applications to that theory.

Date: 2025
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