Douglas Farenick - Extremal matrix states on operator systems

A classical result of Kadison concerning the extension, via the Hahn-Banach theorem, of extremal states on unital selfadjoint linear manifolds (i.e. operator systems) in -algebras is generalised to the setting of noncommutatve convexity, where one has matrix states (i.e. unital completely positive linear maps) and matrix convexity. I will explain here that if is a matrix extreme point of the matrix state space of an operator system R in a unital -algebra A, then has a completely positive extension to a matrix extreme point of the matrix state space of A. This result then leads to a characterisation of extremal matrix states as pure completely positive maps, which is relevant to the recent work of Webster and Winkler on a Krein-Milman theorem in operator convexity.