Completely positive map
In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one that satisfies a stronger, more robust condition.
Definition
Let and be C*-algebras. A linear map is called a 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 then say is k-positive if is a positive map and completely positive if is k-positive for all k.
Properties
- Positive maps are monotone, i.e. for all self-adjoint elements.
- Since for all self-adjoint elements, every positive map is automatically continuous with respect to the C*-norms and its operator norm equals. A similar 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.
Examples
- Every *-homomorphism is completely positive.
- For every linear 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 is automatically completely positive.
- Given the algebras and of complex-valued continuous functions on compact Hausdorff spaces, 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 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.