Operator Subspaces of L(H) with Induced Matrix Orderings

In this paper, we study (possibly non-self-adjoint) subspaces of L(H) together with the induced partially defined involutions and the sequence {M-n(X)}(n epsilon N) of ordered normed spaces. These are called "non-self-adjoint OSs". Two particular concerns are the abstract characterization of non-self-adjoint OSs and the injectivity in the category of non-self-adjoint OSs (known as "MOS-injectivity"). In order to define MOS-injectivity, we need the notion of "unitalization" and "MOS-subspace". We show that in the case of an operator algebra A (which is a non-self-adjoint OS), its unitalization coincides with another unitalization defined in [3] if and only if a form of Stinespring dilation theorem holds for A. On the other hand, through the study of injective objects, we define "MOS-injective envelopes" and "MOS-C*-envelopes" of non-self-adjoint OSs. It is interesting to note that in the case of a unital operator space V (which is a non-self-adjoint OS), its "MOS-injective envelope" need not coincide with the ordinary injective envelope V-inj (they are the same if V is an operator system), but one can identify V-inj with the MOS-injective envelope of a unital operator system associated with V. Similarly, the "MOS-C*-envelope" of an operator algebra A need not be the same as the ordinary C*-envelope C* (A), but one can recover C* (A) as the MOS-C*-envelope of a unital operator system associated with A.