An element is an order unit if for any there is a real number , such that . If has an order unit then .
The cone is called Archimedian if whenever there exists such that for all . If there is an order unit then is Archimedian if whenever for all .
Let be an ordered vector space. If is Archimedian and contains a distinguished order unit then is called an ordered unit space.
Let be involutive vector spaces. We define an involution on the space of all linear mappings from by , . If moreover are ordered vector spaces, then is positive if and . If and are ordered unit spaces a positve map is called unital if .
Let be an involutive vector space. Then is also an involutive vector space by . is a matrix ordered vector space if there are proper cones for all , such that for all and holds21. This means that is a matrix cone .
Let be matrix ordered vector spaces. A linear mapping
is completely positive if
is positive for all
. A complete order isomorphism22 from to is a completely positive map from
that is bijective, such that the inverse map is completely positive.
The well-known Stinespring theorem for completely positive maps reads [Pau86, Theorem 4.1]:
Let be a unital -algebra and let be a Hilbert space. If
is completely positive then there are a Hilbert space
, a unital -homomorphism
and a linear mapping
, such that
for all .
Let be an involutive vector space. Then is called an operator system if it is a matrix ordered ordered unit space, such that
is Archimedian for all
. In this case is an ordered unit space with order unit
for all
, where
is the distinguished order unit of and
is the unit of
.
Example
Let be a Hilbert space. Then, obviously, is an ordered unit space with order unit the identity operator. Using the identification we let . So we see that is an operator system.
Let be an operator system. Then any subspace that is selfadjoint, i.e. , and contains the order unit of is again an operator system with the induced matrix order. So unital -algebras and selfadjoint subspaces of unital -algebras containig the identity are operator systems.
Note that a unital complete order isomorphism between unital -algebras must be a -isomorphism [Cho74, Corollary 3.2]. So unital -algebras are completely characterized by their matrix order. They are not characterized by their order. For instance take the opposite algebra of a unital -algebra . Then but and are not -isomorphic. Obviously .