  Trace inequalities are general techniques with many applications in quantum
information theory, often replacing classical functional calculus in
noncommutative settings. The physics of quantum field theory and holography,
however, motivate entropy inequalities in type III von Neumann algebras that
lack a semifinite trace. The Haagerup and Kosaki $L_p$ spaces enable
re-expressing trace inequalities in non-tracial von Neumann algebras. In
particular, we show this for the generalized Araki-Lieb-Thirring and
Golden-Thompson inequalities from (Sutter, Berta \& Tomamichel 2017). Then,
using the Haagerup approximation method, we prove a general von Neumann algebra
version of univeral recovery map corrections to the data processing inequality
for relative entropy. We also show subharmonicity of a logarithmic p-fidelity
of recovery. Furthermore, we prove that non-decrease of relative entropy is
equivalent to existence of an $L_1$-isometry implementing the channel on both
input states.
