Entropy inequalities have broad applications in theoretical physics and quantum information, as well as mathematical interest. Particularly fundamental is the data processing inequality for relative entropy proven by Lindblad, closely related to the strong subadditivity of quantum entropy shown by Lieb and Ruskai. Several recent results have derived corrections to the data processing inequality via recovery maps. While Petz showed strong subadditivity for relative entropies with respect to C* subalgebras in commuting square, another line of inquiry has noted the value of similar inequalities for algebras not in commuting square. In this talk I will discuss our recent work on such inequalities, as well as work with Marius Junge on recovery in von Neumann algebras. In the latter, we extend the data processing inequality to recently proposed class of fidelity measures, which quantify similarity between states.