IUMJ

Title: Operator subspaces of $\mathcal{L}(H)$ with induced matrix orderings

Authors: Chi-Keung Ng

Issue: Volume 60 (2011), Issue 2, 577-610

Abstract:

In this paper, we study (possibly non-self-adjoint) subspaces of $\mathcal{L}(H)$ together with the induced partially defined involutions and the sequence $\{M_n(X)\}_{n\in\mathbb{N}}$ 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 [D.P. Blecher and C. Le Merdy, \textit{Operator Algebras and Their Modules---an Operator Space Approach}, London Mathematical Society Monographs, New Series, vol. 30, The Clarendon Press, Oxford University Press, Oxford, 2004] 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 $V_{\mathrm{inj}}$ (they are the same if $V$ is an operator system), but one can identify $V_{\mathrm{inj}}$ 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 $A$.