Finite Dimensional Operator Systems, Completely Positive Maps, and Majorization

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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.**