  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.
