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. :-)