On boundary representations

TL;DR

This paper investigates boundary representations in operator systems, demonstrating through explicit examples how certain extreme states lead to boundary representations, highlighting nuanced behaviors in the structure of state extensions.

Contribution

It provides an explicit example showing that only one of the extreme points of a face of state extensions yields a boundary representation, revealing subtle distinctions in the theory.

Findings

An explicit example where only one extreme point yields a boundary representation

Demonstration that faces of state extensions can be intervals with selective boundary representations

Insight into the structure of GNS representations related to boundary representations

Abstract

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.