On boundary representations

Let $S$ be an operator system sitting in its C*-envelope $C^*_{\mathrm{min}}(S)$. Starting with a pure state on $S$, let $F$ be the face of state extensions to $C^*_{\mathrm{min}}(S)$. The dilation theorem of Davidson-Kennedy shows that the GNS representations corresponding to some of the extreme states 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.