Real Operator Algebras and Real Completely Isometric Theory

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.