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 invari-
ant 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.