Real Operator Algebras and Real Completely Isometric Theory

TL;DR

This paper advances the theory of real operator algebras by establishing foundational structures like envelopes and M-ideals, extending complex operator space concepts to the real setting.

Contribution

It develops the theory of real operator algebras, including the existence of envelopes and the characterization of M-ideals in real C*-algebras and operator algebras.

Findings

Existence of injective, C*-envelopes, and non-commutative Shilov boundary for real operator spaces.

Characterization of one-sided M-ideals in real C*-algebras.

Development of real one-sided M-ideal theory.

Abstract

This paper is a continuation of the program started by Ruan in 2003, of developing real operator space theory. In particular, we develop the theory of real operator algebras. We also show among other things that the injective envelope, C*-envelope and non-commutative Shilov boundary exist for a real operator space. We develop real one-sided M-ideal theory and characterize one-sided M-ideals in real C*-algebras and real operator algebras with contractive approximate identity.