On $α$-$z$-Rényi divergence in the von Neumann algebra setting

Abstract

We will investigate the $α$-$z$-Rényi divergence in the general von Neumann algebra setting based on Haagerup non-commutative $L^p$-spaces. In particular, we establish almost all its expected properties when $0 < α< 1$ and some of them when $α> 1$. In an appendix we also give an equality condition for generalized Hölder's inequality in Haagerup non-commutative $L^p$-spaces.