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:2111.02903 ) 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.