M $M$ -Ideals in real operator algebras

Abstract

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