THE STRUCTURE OF C-ASTERISK-CONVEX SETS

Compact C-convex subsets of M(n) 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 M(n). 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.