Extremal matrix states on operator systems

A classical result of Kadison concerning the extension, via the Hahn-Banach theorem, of extremal states on unital self-adjoint linear manifolds (that is, operator systems) in C*-algebras is generalised to the setting of noncommutative convexity, where one has matrix states (that is, unital completely positive linear maps) and matrix convexity. It is shown that if phi is a matrix extreme point of the matrix state space of an operator system R in a unital C*-algebra A, then phi has a completely positive extension to a matrix extreme point Phi of the matrix state space of A. This result leads to a characterisation of extremal matrix states as pure completely positive maps and to a new proof of a decomposition of C*-extreme points.