Tolerance relations and quantization

It is well known that "bad" quotient spaces (typically: non-Hausdorff) can be studied by associating to them the groupoid C*-algebra of an equivalence relation, that in the "nice" cases is Morita equivalent to the C*-algebra of continuous functions vanishing at infinity on the quotient space. It was recently proposed in Connes and van Suijlekom (Tolerance relations and operator systems, arXiv:21 11.02903) that a similar procedure for relations that are reflexive and symmetric but fail to be transitive (i.e. tolerance relations) leads to an operator system. In this paper we observe that such an operator system carries a natural product that, although in general non-associative, arises in a number of relevant examples. We relate this product to truncations of (C*-algebras of) topological spaces, in the spirit of D'Andrea et al. (J Geom Phys 82:18-45, 2014), discuss some geometric aspects and a connection with positive operator valued measures.