Injectivity and operator spaces

Let B(H) be the bounded operators on a Hilbert space H. A linear subspace R ? B(H) is said to be an operator system if 1 ?R and R is self-adjoint. Consider the category $\text{b}$ of operator systems and completely positive linear maps. R ∈ $\text{C}$ is said to be injective if given A ? B, A, B ∈ $\text{C}$, each map A → R extends to B. Then each injective operator system is isomorphic to a conditionally complete C¹-algebra. Injective von Neumann algebras R are characterized by any one of the following: (1) a relative interpolation property, (2) a finite “projectivity” property, (3) letting M_m = B(C^m), each map R → N ? M_m has approximate factorizations R → M_n → N, (4) letting K be the orthogonal complement of an operator system N ? M_m, each map $\text{M}{}_{\mathrm{m}}\text{K}\text{→ R}$ has approximate factorizations $\text{M}{}_{\mathrm{m}}\text{K}\text{→ M}{}_{\mathrm{n}}\text{→ R}$. Analogous characterizations are found for certain classes of C¹-algebras.