Injectivity and operator spaces

Abstract

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 ϵ R and R is self-adjoint. Consider the category b of operator systems and completely positive linear maps. R ∈ C is said to be injective if given A ⊆ B, A, B ∈ C , each map A → R extends to B. Then each injective operator system is isomorphic to a conditionally complete C ∗-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 M m K → R has approximate factorizations M m K → M n → R . Analogous characterizations are found for certain classes of C ∗-algebras.