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 A and B be C*-algebras. A linear map ϕ : A → B is called positive map if ϕ maps positive elements to positive elements: a ≥ 0 ⟹ ϕ ( a ) ≥ 0 .
Any linear map ϕ : A → B induces another map
id ⊗ ϕ : C k × k ⊗ A → C k × k ⊗ B in a natural way. If C k × k ⊗ A is identified with the C*-algebra A k × k of k × k -matrices with entries in A , then id ⊗ ϕ acts as
( a 11 ⋯ a 1 k ⋮ ⋱ ⋮ a k 1 ⋯ a k k ) ↦ ( ϕ ( a 11 ) ⋯ ϕ ( a 1 k ) ⋮ ⋱ ⋮ ϕ ( a k 1 ) ⋯ ϕ ( a k k ) ) . We say that ϕ is k-positive if id C k × k ⊗ ϕ is a positive map, and ϕ is called completely positive if ϕ is k-positive for all k.
Positive maps are monotone, i.e. a 1 ≤ a 2 ⟹ ϕ ( a 1 ) ≤ ϕ ( a 2 ) for all self-adjoint elements a 1 , a 2 ∈ A s a .Since − ∥ a ∥ A 1 A ≤ a ≤ ∥ a ∥ A 1 A every positive map is automatically continuous w.r.t. the C*-norms and its operator norm equals ∥ ϕ ( 1 A ) ∥ B . A similary statement with approximate units holds for non-unital algebras.The set of positive functionals → C is the dual cone of the cone of positive elements of A .Every *-homomorphism is completely positive.For every operator V : H 1 → H 2 between Hilbert spaces, the map L ( H 1 ) → L ( H 2 ) , A ↦ V A V ∗ is completely positive. Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.Every positive functional ϕ : A → C (in particular every state) is automatically completely positive.Every positive map C ( X ) → C ( Y ) 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 C n × n . The following is a positive matrix in C 2 × 2 ⊗ C 2 × 2 :The image of this matrix under I 2 ⊗ T is
[ ( 1 0 0 0 ) T ( 0 1 0 0 ) T ( 0 0 1 0 ) T ( 0 0 0 1 ) T ] = [ 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 ] , 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.
