An inequality related to uncertainty principle in von Neumann algebras

Recently Kosaki proved an inequality for matrices that can be seen as a kind of new uncertainty principle. Independently, the same result was proved by Yanagi et al. The new bound is given in terms of Wigner-Yanase-Dyson informations. Kosaki himself asked if this inequality can be proved in the setting of von Neumann algebras. In this paper we provide a positive answer to that question and moreover we show how the inequality can be generalized to an arbitrary operator monotone function.