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.