  We study the $\alpha$-$z$-R\'enyi divergences $D_{\alpha,z}(\psi\|\varphi)$
where $\alpha,z>0$ ($\alpha\ne1$) for normal positive functionals
$\psi,\varphi$ on general von Neumann algebras, introduced in [S.~Kato and
Y.~Ueda, arXiv:2307.01790] and [S.~Kato, arXiv:2311.01748]. We prove the
variational expressions and the data processing inequality (DPI) for the
$\alpha$-$z$-R\'enyi divergences. We establish the sufficiency theorem for
$D_{\alpha,z}(\psi\|\varphi)$, saying that for $(\alpha,z)$ inside the DPI
bounds, the equality
$D_{\alpha,z}(\psi\circ\gamma\|\varphi\circ\gamma)=D_{\alpha,z}(\psi\|\varphi)<\infty$
in the DPI under a quantum channel (or a normal $2$-positive unital map)
$\gamma$ implies the reversibility of $\gamma$ with respect to $\psi,\varphi$.
Moreover, we show the monotonicity properties of $D_{\alpha,z}(\psi\|\varphi)$
in the parameters $\alpha,z$ and their limits to the normalized relative
entropy as $\alpha\nearrow1$ and $\alpha\searrow1$.
