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
.