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

We will investigate the $\alpha$-$z$-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 $0<\alpha <1$ and some of them when $\alpha >1$. In an appendix we also give an equality condition for generalized Hölder’s inequality in Haagerup non-commutative $L^p$-spaces.