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 (in particular every state) 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 âÂÂ1. Moreover, the eigenvalues of this matrix are 1,1,1 and âÂÂ1. (This matrix happens to be the Choi matrix of T, in fact.) Incidentally, a map æ is said to be co-positive if the composition æ T is positive. The transposition map itself is a co-positive map.
See also
References