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I am trying to grok matrix ordered operator algebras, but I am having a hard time understanding their significance from the definition. Here is the definition (or at least, one way of stating it):

Let $B$ be some complex vector space and $M_n(B)$ the space of $n \times n$ matrixes with entries in $B$; then $B$ is said to be matrix ordered if for every $n\ge1$ there exists a proper cone $M_n(B)_+$ (which is a subset of $M_n(B)_h$ (where $_h$ means hermitean) that is closed under addition and multiplication by positive scalars and furthermore has the property that its intersection with its reflection is exactly $\{0\}$), and furthermore for every $m,n > 0$, $X \in M_n(B)$, and $V \in M_{nm}(B)$ (the set of $n \times m$ matrices) we have that $V^*BV\in M_m(B)$.

I understand what all the parts mean, but I am having trouble getting an intuitive understanding of the whole. If someone here could explain to me why the objects described in the above definition are significant, and possible even give me an example of one, then I would be most grateful. :-)

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Okay. What you have defined is a matrix ordered vector space, not a matrix ordered operator algebra. But this is actually a better place to start.

First question, give examples of such things. The standard example is to let $B$ be a linear subspace of some $M_k(\mathbb{C})$, or more generally, a linear subspace of some $\mathcal{B}(H)$, the bounded operators on some Hilbert space. The point is that we not only have a natural order on $M_k(\mathbb{C})$ --- a matrix is "positive" if it is positive semidefinite --- we also have a natural order on the $n\times n$ matrices over this space, via the natural identification of $M_n(M_k(\mathbb{C}))$ with $M_{nk}(\mathbb{C})$. So $M_n(B)$ inherits this order, for every $n$. Every subspace of $M_k(\mathbb{C})$ not only inherits an order from $M_k(\mathbb{C})$, its tensor product with $M_n(\mathbb{C})$ also inherits an order from $M_{nk}(\mathbb{C})$, for every $n$.

(Incidentally, the order I just described on $B$ might not be interesting because the positive cone might be trivial. Usually one is interested in operator systems, linear subspaces of $M_k(\mathbb{C})$ (or more generally of $\mathcal{B}(H)$) which contain the identity matrix and are stable under the adjoint operation.)

Now why is this significant. The story here is that matrix ordered spaces are important because we care about the natural morphisms between them, namely, linear maps $\phi: B \to B'$ with the property that the natural inflation $\phi_n: M_n(B) \to M_n(B')$ is positive for every $n$ (completely positive linear maps). For motivation about why we care about positivity at matrix levels, I cannot do better than direct you to Stinespring's theorem, one of the first uses of this condition. The Wikipedia page has a sketch of the proof that shows very clearly why complete positivity is the crucial assumption. So, we care about matrix ordered vector spaces because we care about completely positive maps. (Ed Effros made the amusing observation that this is a rare example of a category whose morphisms were discovered before its objects were.)

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