$M$-ideals, yet again: the case of real JB${}^{\ast }$-triples

Abstract

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