Real operator algebras and real completely isometric theory

Abstract

This paper is a continuation of the program started by Ruan (Acta Math Sin (Engl Ser) 19(3):485–496, 2003, Illinois J Math 47(4):1047–1062, 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.