C⁎-extreme entanglement breaking maps on operator systems.

Let E denote the set of all unital entanglement breaking (UEB) linear maps defined on an operator system S ⊂ M d and, mapping into M n. As it turns out, the set E is not only convex in the classical sense but also in a quantum sense, namely it is C ⁎ -convex. The main objective of this article is to describe the C ⁎ -extreme points of this set E. By observing that every EB map defined on the operator system S dilates to a positive map with commutative range and also extends to an EB map on M d , we show that the C ⁎ -extreme points of the set E are precisely the UEB maps that are maximal in the sense of Arveson ([1] and [2]) and that they are also exactly the linear extreme points of the set E with commutative range. We also determine their explicit structure, thereby obtaining operator system generalizations of the analogous structure theorem and the Krein-Milman type theorem given in [8]. As a consequence, we show that C ⁎ -extreme (UEB) maps in E extend to C ⁎ -extreme UEB maps on the full algebra. Finally, we obtain an improved version of the main result in [8] , which contains various characterizations of C ⁎ -extreme UEB maps between the algebras M d and M n. [ABSTRACT FROM AUTHOR]