Grant: Discovery Grants Program - Individual

Operator algebras are the response by mathematicians (mostly John von Neumann, initially) to the development of quantum mechanics. Physicists found themselves using ""mathematical"" objects that somehow were natural but that did not make sense from the point of view of the then known mathematics. The algebras studied by von Neumann did not satisfy the initial goal of setting the framework for quantum mechanics, but surprisingly they achieved way more than that. During the last 50 years operator algebras have provided insight into areas as varied as quantum field theory, knot-theory, quantum computing, and logic, among many others. All the nontrivial examples of operator algebras are infinite-dimensional, and thus highly non-intuitive. Throughout the years researchers have used many tricks and points of view to understand these huge objects. One of these points of view is that of studying their enveloping structures: sometimes an object cannot be looked at explicitly, but it can be seen within a bigger object that has better properties. For C*-algebras, some of these objects are the multiplier algebra, the local multiplier algebra, the regular montone completion, and the injective envelope. My research program investigates these last three enveloping structures. A second branch in my research program is the study of the Schur-Horn theorem. This theorem is a well-understood result for matrices, but its generalizations to operator algebras have proven to be nontrivial. A simple way to state the Schur-Horn theorem is that it characterizes the diagonals of self-adjoint matrices. A corresponding result for normal operators (even finite-dimensional!) is an open question. This kind of question fits nicely in my program of studying joint majorization of commuting families of self-adjoint operators in II_1 factors.