TY  - JOUR
AB  - © 2018, Springer International Publishing AG, part of Springer Nature. We show that Araki and Masuda?s weighted non-commutative vector-valued Lp-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 ?=p2. 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 ?. We thereby also give new proofs for the corresponding finite-dimensional properties. We discuss the limiting cases ??{12,1,?} 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.
AU  - Berta, M
AU  - Scholz, VB
AU  - Tomamichel, M
DA  - 2018/06/01
DO  - 10.1007/s00023-018-0670-x
EP  - 1867
JO  - Annales Henri Poincare
PY  - 2018/06/01
SP  - 1843
TI  - Rényi Divergences as Weighted Non-commutative Vector-Valued L<inf>p</inf> -Spaces
VL  - 19
Y1  - 2018/06/01
Y2  - 2026/05/28
ER  -