Completely positive map

Short description: C*-algebra mapping preserving positive elements

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.

## Contents

* 1 Definition
* 2 Properties
* 3 Examples
* 4 See also

## Definition

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:ℂk×k⊗A->ℂk×k⊗B

in a natural way. If ℂk×k⊗A is identified with the C*-algebra Ak×k of k×k-matrices with entries in A, then id⊗ϕ acts as

(a11⋯a1k⋮⋱⋮ak1⋯akk)↦(ϕ(a11)⋯ϕ(a1k)⋮⋱⋮ϕ(ak1)⋯ϕ(akk)).

We say that ϕ is k-positive if idℂk×k⊗ϕ is a positive map, and ϕ is called completely positive if ϕ is k-positive for all k.

## Properties

* Positive maps are monotone, i.e. a1≤a2⟹ϕ(a1)≤ϕ(a2) for all self-adjoint elements a1,a2∈Asa.
* Since −‖a‖A1A≤a≤‖a‖A1A every positive map is automatically continuous with respect to the C*-norms and its operator norm equals ‖ϕ(1A)‖B. A similar 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.

## Examples

* Every *-homomorphism is completely positive.
* For every linear operator V:H1->H2 between Hilbert spaces, the map L(H1)->L(H2), A↦VAV∗ 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 ℂn×n. The following is a positive matrix in ℂ2×2⊗ℂ2×2:

[(1000)(0100)(0010)(0001)]=[1001000000001001].

The image of this matrix under I2⊗T is

[(1000)T(0100)T(0010)T(0001)T]=[1000001001000001],

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.

## See also

* Choi's theorem on completely positive maps

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