In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition.
Let and be C*-algebras. A linear map is called positive map if maps positive elements to positive elements: .
Any linear map induces another map
in a natural way. If is identified with the C*-algebra of -matrices with entries in , then acts as
We say that is k-positive if is a positive map, and is called completely positive if is k-positive for all k.
- Positive maps are monotone, i.e. for all self-adjoint elements .
- Since every positive map is automatically continuous w.r.t. the C*-norms and its operator norm equals . A similary statement with approximate units holds for non-unital algebras.
- The set of positive functionals is the dual cone of the cone of positive elements of .
- Every *-homomorphism is completely positive.
- For every operator between Hilbert spaces, the map is completely positive. Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
- Every positive functional (in particular every state) is automatically completely positive.
- Every positive map is completely positive.
- The transposition of matrices is a standard example of a positive map that fails to be 2-positive. Let T denote this map on . The following is a positive matrix in :
The image of this matrix under is
- which is clearly not positive, having determinant -1. Moreover, the eigenvalues of this matrix are 1,1,1 and -1.
- Incidentally, a map Φ is said to be co-positive if the composition Φ T is positive. The transposition map itself is a co-positive map.