  We propose an extension of the sandwiched R\'enyi relative $\alpha$-entropy
to normal positive functionals on arbitrary von Neumann algebras, for the
values $\alpha>1$. For this, we use Kosaki's definition of noncommutative
$L_p$-spaces with respect to a state. We show that these extensions coincide
with the previously defined Araki-Masuda divergences [M. Berta et al., Annales
Henri Poincar\'e, 19:1843--1867, 2018] and prove some of their properties, in
particular the data processing inequality with respect to positive normal
unital maps. As a consequence, we obtain monotonicity of the Araki relative
entropy with respect to such maps, extending the results of [A. M\"uller-Hermes
and D. Reeb. Annales Henri Poincar\'e,18:1777--1788, 2017] to arbitrary von
Neumann algebras. It is also shown that equality in data processing inequality
characterizes sufficiency (reversibility) of quantum channels.
