On $\alpha$-$z$-R\'{e}nyi divergence in the von Neumann algebra setting

Abstract

We will investigate the $\alpha$-$z$-R\'{e}nyi 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 $0 < \alpha < 1$ and some of them when $\alpha > 1$. In an appendix we also give an equality condition for generalized H\"{o}lder's inequality in Haagerup non-commutative $L^p$-spaces.