We study an extension of the sandwiched Rényi 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 Jenčová (Ann Henri Poincaré 19:2513–2542, 2018), where the values \(\alpha >1\) were studied. We use the Araki–Masuda divergences of Berta et al. (Ann Henri Poincaré 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 \(\alpha \in [1/2,1)\), 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.

我们研究了冯诺依曼代数上正常正泛函的夹心 Rényi 相对熵的扩展，对于参数值\(\alpha \in [1/2,1)\)。这项工作旨在作为 Jenčová (Ann Henri Poincaré 19:2513–2542, 2018) 的延续，其中研究了值\(\alpha >1\)。我们使用 Berta 等人的 Araki-Masuda 散度。(Ann Henri Poincaré 9:1843–1867, 2018) 并在 Kosaki 的非交换\(L_p\)空间的框架内处理它们。使用 F. Hiai 最近获得的变分公式，对于\(\alpha \in [1/2,1)\)，我们证明了关于保留正迹图的数据处理不等式，并证明对于\(\alpha \in (1/2,1)\)，等式表征了任何 2-positive 迹保留映射的充分性（可逆性）。