  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.
