Talk at IWOTA 2023

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 the relation

d(x, y) < ϵ 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) < ϵ on

a path metric measure space. (joint work with Alain Connes)