A $W*$-algebraic formalism for Information Geometry

What is Information Geometry? A typical answer to this question is that Information Geometry is the application of differential geometric methods to the investigation of Information Theory. This leads, depending on the specific information-theoretical task one is considering, to the study of a seemingly quite inhomogeneous collection of Riemannian structures. Some typical examples being the interior of a simplex equipped with the Fisher-Rao metric, or the Bloch ball equipped with the Bures-Helstrom metric. A question that is then quite natural to ask is whether these structures have some common features, or if they can be seen as different instances of the same general structure. As we will see in this talk, following this aim will bring us to construct a formalism for Information Geometry that has at its center the notion of $W*$-algebras.