Tolerance relations and operator systems

TL;DR

This paper generalizes noncommutative geometry to tolerance relations, introducing operator system invariants like the $C^*$-envelope and analyzing their structure and state spaces, especially for metric space relations.

Contribution

It extends noncommutative geometric methods to tolerance relations, providing new invariants and a detailed analysis of associated operator systems and state spaces.

Findings

Defined operator systems for tolerance relations

Identified new invariants such as the $C^*$-envelope

Analyzed pure state spaces for metric space relations

Abstract

We extend the scope of noncommutative geometry by generalizing the construction of the noncommutative algebra of a quotient space to situations in which one is no longer dealing with an equivalence relation. For these so-called tolerance relations, passing to the associated equivalence relation looses crucial information as is clear from the examples such as coarse graining in physics or the relation $d(x,y)< \varepsilon$ on a metric space. Fortunately, thanks to the formalism of operator systems such an extension is possible and provides new invariants, such as the $C^*$-envelope and the propagation number. After a thorough investigation of the structure of the (non-unital) operator systems associated to tolerance relations, we analyze the corresponding state spaces. In particular, we determine the pure state space associated to the operator system for the relation $d(x,y)< \varepsilon$ on a path metric measure space.