On boundary representations

Let S be an operator system sitting in its C∗-envelope C∗min(S). Starting with a pure state on S, let F be the face of state extensions to C∗min(S). The dilation theorem of Davidson-Kennedy shows that the GNS representations corresponding to some of the extreme points of F are boundary representations. We construct an explicit example in which F is an interval and only one of the two extreme points yields a boundary representation.