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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.

## Definition

Let and be C*-algebras. A linear map is called **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 say that is **k-positive** if is a positive map, and is called **completely positive** if is k-positive for all k.

## 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 (in particular every state) 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 :

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.