K\"ahler-Poisson algebras

We introduce K\"ahler-Poisson algebras as analogues of algebras of smooth functions on K\"ahler manifolds, and prove that they share several properties with their classical counterparts on an algebraic level. For instance, the module of inner derivations of a K\"ahler-Poisson algebra is a finitely generated projective module, and allows for a unique metric and torsion-free connection whose curvature enjoys all the classical symmetries. Moreover, starting from a large class of Poisson algebras, we show that every algebra has an associated K\"ahler-Poisson algebra constructed as a localization. At the end, detailed examples are provided in order to illustrate the novel concepts.