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 A and B be C*-algebras. A linear map is called a positive map if \phi maps positive elements to positive elements:. Any linear map \phi:A\to B induces another map in a natural way. If is identified with the C*-algebra of k\times k-matrices with entries in A, then acts as \phi is called k-positive if is a positive map and completely positive if \phi is k-positive for all k.

Properties

Examples

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. (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 \circ T is positive. The transposition map itself is a co-positive map.