In this talk, I will introduce the concept of order bounds and order bounded maps on ordered vector spaces over the real numbers. I will use these concepts to characterize self-adjoint subspaces of C*-algebras. As an application, I will demonstrate necessary and sufficient conditions under which a completely positive completely contractive map from a self-adjoint operator space to B(H) has a completely positive completely contractive lift, generalizing the well-known Arveson extension theorem.