Noncommutative geometry and operator systems

We give an overview of the recent interactions between noncommutative geometry and operator systems. We will see that the structure of an operator system is the minimal structure required to be able to speak of positive elements, states, pure states, etc. After presenting the general theory, we will illustrate this by many examples, ranging from spectral truncations of geometric spaces, to metric spaces up to a finite resolution. We will also present a general approach to analyzing (Gromov-Hausdorff) convergence results, which we will illustrate once again in these examples.