  We show that Araki and Masuda's weighted non-commutative vector valued
$L_p$-spaces [Araki \& Masuda, Publ. Res. Inst. Math. Sci., 18:339 (1982)]
correspond to an algebraic generalization of the sandwiched R\'enyi divergences
with parameter $\alpha = \frac{p}{2}$. 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
$\alpha$. We thereby also give new proofs for the corresponding
finite-dimensional properties. We discuss the limiting cases $\alpha\to
\{\frac{1}{2},1,\infty\}$ 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 $L_p$-spaces and an Araki-Lieb-Thirring inequality for
states on von Neumann algebras.
