Images of Order Zero Maps

We give sufficient conditions allowing one to build a C*-algebraic structure on a self-adjoint linear subspace of a C*-algebra in such a way that the subspace is naturally identified with the resulting C*-algebra via a completely positive order zero map. This leads to necessary and sufficient conditions to realize a self-adjoint linear subspace of a C*-algebra as the image of an order zero map from a C*-algebra and to determine whether it is completely order isomorphic to a C*-algebra via an order zero map.