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Extremal Matrix States on Operator Systems
Author(s) -
Farenick Douglas R.
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610799008613
Subject(s) - noncommutative geometry , mathematics , matrix (chemical analysis) , extreme point , pure mathematics , operator (biology) , state (computer science) , nonnegative matrix , discrete mathematics , combinatorics , symmetric matrix , quantum mechanics , physics , biochemistry , eigenvalues and eigenvectors , repressor , transcription factor , composite material , gene , algorithm , chemistry , materials science
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.