The facialQ-topology for compact convex sets

The structure of the set of closed two-sided ideals in aC*-algebraU with identity is described by means of a topology on the set _eK of extreme points of the state spaceK ofU. Recent results of Alfsen, Andersen, Combes, Perdrizet, Wils, and others have shown that such a topology can be defined on the set _eK of extreme points of an arbitrary compact convex subset of a locally convex Hausdorff topological vector space.The structure of the set of closed left ideals in aC*-algebraU with identity can also be described by means of a set of subsets of the set _eK of extreme points of its state spaceK. Akemann, Giles, and Kummer showed that this formed a more general structure than a topology which was called aq-topology. In this paper it is shown that for a reasonably wide class of compact convex subsetsK of locally convex Hausdorff topological vector spaces such aq-topology can also be defined on _eK and that it shares many of the properties of theq-topology defined forC*-algebras. The methods used depend strongly upon recent results of Alfsen and Shultz on the spectral theory of affine functions on compact convex sets.