Talk given at Corfu Summer Institute (21 September 2022) [slides]
Abstract: We extend the traditional framework of noncommutative geometry in order to deal with spectral truncations of geometric spaces. In our approach the traditional role played by C*-algebras is taken over by so-called operator systems. We study the convergence aspects and find general conditions on sequences of operator system spectral triples that allows one to prove a result on Gromov-Hausdorff convergence of the corresponding state spaces when equipped with Connes’distance formula. We exemplify this result for spectral truncations of the circle.