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 φ is a matrix extreme point of the matrix state space of an operator system R in a unital C *‐algebra A , then φ has a completely positive extension to a matrix extreme point Φ 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.