Completely positive map

Definition

Properties

• Positive maps are monotone, i.e. for all self-adjoint elements.
• Since 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.
• 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 :