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M-ideals in real operator algebras

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

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.

M-ideals in real operator algebras

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.
Paper Structure (4 sections, 17 theorems, 20 equations)

This paper contains 4 sections, 17 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on real Jordan operator algebras
  3. M-ideals in operator algebras
  4. One-sided $M$-ideals

Key Result

Theorem 2.1

Let $A$ be an approximately unital real Jordan operator algebra and let $B = C^*_e(A)$, with $A$ considered as a Jordan subalgebra. Then ${\rm LM}(A) \cong \{\eta\in B^{**} : \eta A\subset A \}$. This is completely isometrically isomorphic to the (associative) operator algebra ${\mathcal{M}}_\ell(A

Theorems & Definitions (31)

  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Corollary 3.3
  • ...and 21 more