Tolerance relations and operator systems

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)