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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

Examples

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. (This matrix happens to be the Choi matrix of T, in fact.)
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