Noncommutative spaces at finite resolution

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. 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.