Abstract: We extend the traditional framework of noncommutative geometry in order to deal with two types of approximation of metric spaces. On the one hand, we consider spectral truncations of geometric spaces, while on the other hand, we consider metric spaces up to a finite resolution. In our approach the traditional role played by C*-algebras is taken over by so-called operator systems. Essentially, this is the minimal structure required on a space of operators to be able to speak of positive elements, states, pure states, etc. We consider C*-envelopes and introduce a propagation number for operator systems, which we show to be an invariant under stable equivalence and use it to compare approximations of the same space.

We illustrate our methods for concrete examples obtained by spectral truncations of the circle, and of metric spaces up to finite resolution. The first are operator systems of finite-dimensional Toeplitz matrices, the second are suitable subspaces of the compact operators. We also analyze the cones of positive elements and the pure state spaces for these operator systems, which turn out to possess a very rich structure.