Order Embedding of a Matrix Ordered Space

We characterize certain properties in a matrix ordered space in order to embed it in a C * -algebra. Let such spaces be called C * -ordered operator spaces. We show that for every self-adjoint operator space there exists a matrix order (on it) to make it a C * -ordered operator space. However, the operator space dual of a (nontrivial) C * -ordered operator space cannot be embedded in any C * -algebra.