M-ideals, yet again: the case of real \(\hbox {JB}^*\)-triples

We prove that a subspace of a real \(\hbox {JBW}^*\) ${\text{JBW}}^{\ast }$ -triple is an M-summand if and only if it is a \(\hbox {weak}^*\) ${\text{weak}}^{\ast }$ -closed triple ideal. As a consequence, M-ideals of real \(\hbox {JB}^*\) ${\text{JB}}^{\ast }$ -triples correspond to norm-closed triple ideals. As in the setting of complex \(\hbox {JB}^*\) ${\text{JB}}^{\ast }$ -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 \(\hbox {C}^*\) ${\text{C}}^{\ast }$ -algebras and \(\hbox {JB}^*\) ${\text{JB}}^{\ast }$ -triples. The results in this note also provide a full characterization of all M-ideals in real \(\hbox {C}^*\) ${\text{C}}^{\ast }$ -algebras, real \(\hbox {JB}^*\) ${\text{JB}}^{\ast }$ -algebras and real TROs.