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Journal of the London Mathematical Society 2000 61(3):885-892; doi:10.1112/S0024610799008613
© 2000 by London Mathematical Society
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© The London Mathematical Society

Extremal Matrix States on Operator Systems

Douglas R. Farenick

Department of Mathematics and Statistics, University of Regina Regina, Saskatchewan S4S 0A2, Canada

Received 15 February 1999.

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 {varphi} is a matrix extreme point of the matrix state space of an operator system R in a unital C*-algebra A, then {varphi} 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.


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