Journal of Operator Theory

Volume 94, Issue 1, Summer 2025 pp. 23-34.

On boundary representations

Authors: Kenneth R. Davidson (1), Michael Hartz (2)
Author institution: (1) Pure Mathematics Department, University of Waterloo, Waterloo, ON N2L 3G1, Canada
(2) Fachrichtung Mathematik, Universitaet des Saarlandes, 66123 Saarbruecken, Germany

Summary: 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 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.

DOI: http://dx.doi.org/10.7900/jot.2023aug08.2439
Keywords: boundary representation, pure state, GNS representation

Contents Full-Text PDF