positiemaps

Positiemaps are a term used in some mathematical contexts to denote positivity-preserving linear maps between ordered vector spaces, especially in the setting of operator algebras and matrix algebras. Formally, let A and B be C*-algebras, or more generally ordered vector spaces equipped with a positive cone. A linear map f: A -> B is called a positimap if it maps positive elements to positive elements, that is, f(A_+) ⊆ B_+. In finite-dimensional matrix algebras, this means f sends positive semidefinite matrices to positive semidefinite matrices.

Key properties include that the composition of positiemaps is a positimap, and both the identity map and the zero map are positiemaps. Positivity alone does not imply a stronger notion called complete positivity. A classical example is the transpose map on matrix algebras: it is positive since it preserves positive semidefinite matrices, but it is not completely positive when the matrix size n exceeds 1.

Positiemaps are closely related to completely positive maps. A map f is completely positive if id_k ⊗ f is positive for all k, which strengthens the notion of positivity and is central in quantum information theory. Completely positive, trace-preserving maps model quantum channels, whereas merely positive maps may fail to remain positive when extended.

Examples common in practice include the identity map on any C*-algebra, and the trace map from a matrix algebra to the complex numbers, both of which are positive. In probabilistic and functional-analytic contexts, certain Markov operators and expectations can be viewed as positiemaps, acting on spaces of functions or operators while preserving positivity.

See also: Positive linear functional, positive map, completely positive map, Markov operator, C*-algebra, ordered vector space.