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Inequalities in Operator Algebras [MA5930]

Summersemester 2019

Professor Eric A. Carlen, John von Neumann Gastprofessor

Lecturer: Professor Eric A. Carlen


Inequalities in Operator Algebras

Functional inequalities have long played an important role in many fields of science, especially statistical physics and information theory. In these two fields, there are many inequalities associated with entropy that are fundamental to the foundations of these subjects. Naturally, as quantum statistical mechanics, and later, quantum information theory, began to be developed, the mathematical investigation of related inequalities in the quantum setting began to be developed. This is an ongoing and active area of research. Many new phenomena arise in the quantum setting, and many of the methods used to prove inequalities in the classical setting for probability densities fail in the quantum setting for non-commuting density matrices. For example, in the classical setting, many proofs rely on convexity arguments. However, the appropriate non-commutative analog, operator convexity, is much more restrictive, and many functions that are convex are not operator convex.

The proper setting for the development of this theory is in the context of operator algebras, especially von Neumann algebras. Even if one is only concerned with operators that are finite dimensional matrices, as is often, but not always, the case in quantum information theory, ideas that were introduced in the development of the theory of von Neumann algebras have proved very useful. For operators on a finite-dimensional or separable Hilbert space, many of the inequalities we shall consider involve the trace, and they may be viewed as trace inequalities. For example, the von Neumann entropy is defined in terms of the trace, and any inequality for the von Neumann entropy is a trace inequality in this sense. One of the most fundamental inequalities with which we shall be concerned is the Lieb Concavity Theorem, which leads to the Strong Subadditivity of Quantum Entropy, proved by Lieb and Ruskai, and to Lindblad's theorem on the monotonicity of quantum relative entropy. All of these inequalities were first formulated as trace inequalities. Not long after they were proved in the early 1970's, several alternative proofs were provided, with the approaches of Epstein and of Araki being especially influential. Araki not only gave a new proof, but he extended the very notion of relative entropy to a fully general operator algebra setting in which there need not be a trace. In his setting, there is no analog of the von Neumann entropy itself, but there is an analog of the quantum relative entropy, and he proved the monotonicity for it in his fully general setting. His approach made use of the Tomita-Takesaki Theory of von Neumann algebras, to which he made fundamental contributions. It took a number of years before it was realized that his ideas could be very fruitfully applied in the finite dimensional setting too, where the Tomita-Takesaki Theory is much more elementary, but still extremely powerful. Petz was among the first to realize and exploit this. It remains an active branch of this very active field.

The prerequisite for this course is an introductory course in functional analysis. In addition, familiarity with the basic concepts of operator algebras, up to having been through Gelfand-Neumark-Segal Construction, would be helpful. My notes from part of a graduate course on the subject, available on these web pages, cover more than enough background in full detail with complete proofs, and these notes assume only a knowledge of basis functional analysis. Those who are interested in the course, but who have not studied the basic theory of operator algebras, may wish to consult these notes in advance of the course.