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

We study an extension of the sandwiched R'enyi relative entropies for normal positive functionals on a von Neumann algebra, for parameter values $\alpha\in [1/2,1)$. This work is intended as a continuation of [A. Jen\v{c}ov'a, Ann. Henri Poincar'e 19, 2513-2542, 2018], where the values $\alpha >1$ were studied. We use the Araki-Masuda divergences of [Berta et al., Ann. Henri Poincar'e 9, 1843-1867, 2018] and treat them in the framework of Kosaki’s noncommutative $L_p$-spaces. Using the variational formula, recently obtained by F. Hiai, for $\alpha \in [1\mathrm{/}2,1)$, we prove the data processing inequality with respect to positive trace preserving maps and show that for $\alpha\in (1/2,1)$, equality characterizes sufficiency (reversibility) for any 2-positive trace preserving map.