Entropy and its variants play a central role in both classical- and quantum information theory. In last decade, the connection between quantum entropies and noncommutative $L_p$-norms has found many application in quantum information. In this talk, I will explain how this connection provides functional analytic tools to entropic quantity, such as quantum channel capacity and entropic uncertainty principle. A new connection between relative entropy and subfactor index will also be mentioned. This talk is based on joint works with Marius Junge and Nicholas LaRacuente.

Majorization is a concept from linear algebra that is used to compare disorderness in physics, computer science, economics and statistics. Recently, Gour et al (2018) extended matrix majorization to the quantum mechanical setting to accommodate ordering of quantum states. In this talk, I will discuss a generalization of their definition and entropic characterization of quantum majorization to the infinite dimensional setting, using operator space tensor products and duality. This is based on joint work with Li Gao, Satish Pandey and Sarah Plosker.

Let A be a simple unital C*-algebra. Suppose that a finite group G acts on A, and that the action has the weak tracial Rokhlin property, a generalization of the Rokhlin property which uses positive elements instead of projections, and is fairly common. We prove that, after discarding the classes of the nonzero projections, the Cuntz semigroup of the fixed point algebra is just the fixed points in the Cuntz semigroup of A. For context, for algebras without strict comparison, the Cuntz semigroup is often very hard to compute. As a corollary, we prove that the radius of comparison of the crossed product satisfies rc (C^* (G, A)) \leq [1 / card (G)] rc (A). We also give an example of a simple separable unital AH algebra A and an action of the two element group G on A which has the Rokhlin property, and such that rc (A) and rc (C^* (G, A)) are both strictly positive. The way the weak tracial Rokhlin property is used in the proof is different from the usual methods in C*-algebras. Joint work with M. Ali Asadi-Vasfi and Nasser Golestani.