M-ideals in real operator algebras

TL;DR

This paper extends the characterization of M-ideals from real JB*-triples to more general real operator and Jordan operator algebras, providing new insights and simple characterizations of one-sided M-ideals.

Contribution

It generalizes the correspondence between M-ideals and norm-closed triple ideals to real operator algebras and Jordan operator algebras, including non-selfadjoint cases.

Findings

M-ideals in real operator algebras correspond to norm-closed triple ideals.

Characterizations of one-sided M-ideals in real operator algebras.

Extension of previous results from real JB*-triples to broader classes.

Abstract

In a recent paper we showed 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, including real C*-algebras, real JB*-algebras and real TROs, correspond to norm-closed triple ideals. In the present paper we extend this result to (possibly non-selfadjoint) real operator algebras and Jordan operator algebras, where the argument is necessarily different. We also give simple characterizations of one-sided M-ideals in real operator algebras, and give some applications to that theory.