Monotone metrics on matrix spaces

The study of monotone inner products under stochastic mappings on the space of matrices was initiated by Morozova and Chentsov, motivated by information geometry. They did not show a monotone metric, but proposed several candidates. The main result of the present paper is to provide an abundance of monotone metrics by means of operator monotone functions and to characterize them. It turns out that there is a correspondence between monotone metrics and operator means in the sense of Kubo and Ando. It follows that all proposals of Morozova and Chentsov are indeed monotone metrics.