R\'enyi relative entropies and noncommutative $L_p$-spaces

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.