An (not   necessarily unital) operator system is a self-adjoint subspace of , equipped with the induced matrix   norm and the induced matrix cone. We say that an operator system  is dualizable if one can find an equivalent   dual matrix norm on the dual space  such that under this dual matrix norm and   the canonical dual matrix cone,  becomes a dual operator system.
   

We show that   an operator system  is dualizable if and only if the ordered   normed space  satisfies a form of bounded decomposition   property. In this case,

is the   largest dual matrix norm that is equivalent to and dominated by the original   dual matrix norm on  that turns it into a dual operator system,   denoted by . It can be shown that  is again dualizable.

For every   completely bounded completely positive map  between dualizable operator systems, there   is a unique weak-*-continuous completely bounded completely positive map    that is compatible with the dual map . From this, we obtain a full   and faithful functor from the category of dualizable operator systems to that   of dualizable dual operator systems.

Moreover, we   will verify that that if  is either a C*-algebra or a unital operator   system, then  is dualizable and the canonical   weak-*-homeomorphism from the unital operator system  to the operator system  is a completely isometric complete order   isomorphism.
 
  Furthermore, via the duality functor above, the category of C*-algebras and   that of unital operator systems (both equipped with completely positive   complete contractions as their morphisms) can be regarded as full   subcategories of the category of dual operator systems (with   weak-*-continuous completely positive complete contractions as their   morphisms).
 
  Consequently, a nice duality framework for operator systems is obtained,   which includes all C*-algebras and all unital operator systems.

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