We show that Araki and Masuda’s weighted non-commutative vector-valued $$L_p$$ L p -spaces (Araki and Masuda in Publ Res Inst Math Sci Kyoto Univ 18:339–411, 1982) correspond to an algebraic generalization of the sandwiched Rényi divergences with parameter $$\alpha = \frac{p}{2}$$ 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 $$\alpha $$ α . We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases $$\alpha \rightarrow \{\frac{1}{2},1,\infty \}$$ α → { 1 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 $$L_p$$ L p -spaces and an Araki–Lieb–Thirring inequality for states on von Neumann algebras.
Annales Henri Poincaré – Springer Journals
Published: Mar 17, 2018
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