If φ : S → T is a completely positive (cp) linear map of operator systems and if J = ker φ, then the quotient vector space S/J may be endowed with a matricial ordering through which S/J has the structure of an operator system. Furthermore, there is a uniquely determined cp maṗ φ : S/J → T such that φ =φ • q, where q is the canonical linear map of S onto S/J . The cp map φ is called a complete quotient map ifφ is a complete order isomorphism between the operator systems S/J and T . Herein we study certain quotient maps in the cases where S is a full matrix algebra or a full subsystem of tridiagonal matrices.Our study of operator system quotients of matrix algebras and tensor products has applications to operator algebra theory. In particular, we give a new, simple proof of Kirchberg's Theoremshow that an affirmative solution to the Connes Embedding Problem is implied by various matrix-theoretic problems, and give a new characterisation of unital C * -algebras that have the weak expectation property.

We formulate a general framework for the study of operator systems arising from discrete groups. We study in detail the operator system of the free group Fn on n generators, as well as the operator systems of the free products of finitely many copies of the two-element group Z 2 . We examine various tensor products of group operator systems, including the minimal, the maximal, and the commuting tensor products. We introduce a new tensor product in the category of operator systems and formulate necessary and sufficient conditions for its equality to the commuting tensor product in the case of group operator systems. We express various sets of quantum correlations studied in the theoretical physics literature in terms of different tensor products of operator systems of discrete groups. We thus recover earlier results of Tsirelson and formulate a new approach for the study of quantum correlation boxes.2010 Mathematics Subject Classification. Primary 46L06, 46L07; Secondary 46L05, 47L25, 47L90.

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Abstract. In this paper we study the space S H (A) of unital completely positive linear maps from a C * -algebra A to the algebra B(H) of continuous linear operators on a complex Hilbert space H. The state space of A, in this notation, is S C (A). The main focus of our study concerns noncommutative convexity. Specifically, we examine the C * -extreme points of the C * -convex space S H (A). General properties of C * -extreme points are discussed and a complete description of the set of C * -extreme points is given in each of the following cases: (i) the cases S C 2 (A), where A is arbitrary ; (ii) the cases S C r (A), where A is commutative; (iii) the cases S C r (Mn), where Mn is the C * -algebra of n × n complex matrices. An analogue of the Krein-Milman theorem will also be established.

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