Operator systems for tolerance relations on finite sets

TL;DR

This paper investigates the duals of operator systems linked to tolerance relations and graphs on finite sets, providing concrete realizations for chordal graphs and computing key properties like $C^*$-envelopes.

Contribution

It introduces explicit realizations of dual operator systems for chordal graphs and computes their $C^*$-envelopes and extremal properties, advancing understanding of their structure.

Findings

Concrete realizations of duals inside finite-dimensional $C^*$-algebras for chordal graphs

Computed $C^*$-envelopes, propagation numbers, and extremal rays of these duals

Applied results to operator systems of band matrices

Abstract

We study the duals of a certain class of finite-dimensional operator systems, namely the class of operator systems associated to tolerance relations on finite sets or equivalently the class of operator systems that are associated with graphs. In the case where the graphs associated with these operator systems are chordal we are able to find concrete realizations of their duals as sitting inside of finite-dimensional $C^*$-algebras. We then use these concrete realizations to compute the $C^*$-envelopes, propagation numbers and extremal rays of these duals in the chordal case. Finally, we exemplify our results by applying them to operator systems of band matrices.