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.

Compact C*-convex subsets of Mn correspond exactly to n-th matrix ranges of operators. The main result of this paper is to discover the “right” analog of linear extreme points, called structural elements, and then to prove a generalised Krein-Milman theorem for C*-convex subsets of Mn. The relationship between structural elements and an earlier attempted generalisation, called C*-extreme points, is examined, solving affirmatively a conjecture of Loebl and Paulsen [8]. An improved bound for a C* -convex version of the Caratheodory theorem for convex sets is also given.

In the C*-algebra M" of complex nxn matrices, we consider the notion of noncommutative convexity called C*-convexity and the corresponding notion of a C*-extreme point. We prove that each irreducible element of M" is a C""-extreme point of the C*-convex set it generates, and we classify the C*-extreme points of any C*-convex set generated by a compact set of normal matrices.

In the C*-algebra M" of complex nxn matrices, we consider the notion of noncommutative convexity called C*-convexity and the corresponding notion of a C*-extreme point. We prove that each irreducible element of M" is a C""-extreme point of the C*-convex set it generates, and we classify the C*-extreme points of any C*-convex set generated by a compact set of normal matrices.