Real operator algebras and real completely isometric theory

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.