Tolerance relations and operator systems

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) < 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.

Tolerance Relations and Quantization

“…, n} is finite and that the topology on both X and R is the discrete one. The convolution product is then given by (8) in the basis of delta functions. We will denote by A(R) the algebra C c (R) with product (8), and call it the tolerance algebra associated to R.…”

“…. , n}, A = A(R) ⊂ M n (C) the vector space in Example 16, T the map (19), the product (8). There are two natural partial orders on A(R), which we will denote by ≥ and , defined as follows.…”

“…van Suijlekom in [7] that from any tolerance relation one can get an operator system, which they propose as a starting point for a generalization of one of the main ingredients of Noncommutative Geometry, namely spectral triples [5,17,15]. The connection between tolerance relations and operator systems was then developed in [8]. It turns out that these operator systems possess, in fact, the additional structure of an algebra.…”