Rényi divergences as weighted non-commutative vector valued $L_p$-spaces

We show that Araki and Masuda’s weighted non-commutative vector valued Lp-spaces [Araki & Masuda, Publ. Res. Inst. Math. Sci., 18:339 (1982)] correspond to an algebraic generalization of the sandwiched Rényi divergences with parameter α = p 2 . Using complex interpolation theory, we prove various fundamental properties of these divergences in the setup of von Neumann algebras, including a data processing inequality and monotonicity in α. We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases α → { 2 , 1, ∞} leading to minus the logarithm of Uhlmann’s fidelity, Umegaki’s relative entropy, and the max-relative entropy, respectively. As a contribution that might be of independent interest, we derive a Riesz-Thorin theorem for Araki-Masuda Lp-spaces and an Araki-Lieb-Thirring inequality for states on von Neumann algebras.