Finite Dimensional Operator Systems, Completely Positive Maps, and Majorization

点击翻译按钮获取中文摘要

Operator Algebras is an area of Mathematics that grew out of the efforts of mathematicians-pioneered by John von Neumann-to create mathematics that fit what physicists were doing in Quantum Mechanics. Indeed, as it has happened and continues to happen, physicists found themselves using objects that in some sense were mathematical-and that fit their intuition on how their models were working-but did not make sense from the point of view of the accepted mathematics of the time. The mathematical area created by von Neumann did not directly fulfill the goal of becoming the language of Quantum Mechanics, but it became a mathematical world on its own. Over the last 60 years, Operator Algebras have provided insight into areas as diverse as quantum field theory, knot theory, logic, quantum information and quantum computing, among others. The algebras considered by operator algebraists are naturally infinite-dimensional, and so they are not very amenable to our intuition. This has led researchers to, besides developing some intuition, create a myriad of tricks and points of view to understand parts of these immense objects. One of these points of view is that of enveloping structures. Sometimes it is possible to say something about an object by considering it inside a bigger, more tractable object. For C*-algebras, some of these enveloping structures include the double dual, the multiplier algebra, and the injective envelope. My research program investigates these last two objects. For Operator Systems, the most natural enveloping object is the C*-envelope, defined by Arveson in 1972, and this object is also part of my research program. Operator systems are subspaces of operators that contain the identity and the adjoints of all its operators. They are the natural objects on which to study completely positive maps. Even in small dimensions, operators systems are not well-understood, and a classification up to complete order isomorphism is lacking. My program aims to fill this gap, by working towards and effective classification of finite-dimensional operator systems are their C*-envelopes. Another branch of my research program consists of the study of majorization and the Schur-Horn theorem. This theorem is a very well understood result about matrices, such that its generalizations to an infinite-dimensional setting are non-trivial. In slight technical terms, the Schur-Horn theorem characterizes the possible diagonals of a self-adjoint matrix under different choices of an orthonormal basis. Still in finite-dimension, a generalization of this theorem to normal operators is a question no one knows the answer to! My research on commuting families of selfadjoint operators provides a context where this may be studied successfully. Majorization appears naturally in Quantum Information, and my program also investigates this connection, in particular with the so-called trumping majorization.