Semialgebraic Operator Algebra With Applications

Semialgebraic geometry studies sets of points defined by polynomial inequalities. The non- commutative version replaces points by matrices, and thus gives rise to sets of matrices of all sizes simultaneously. Operator algebra, on the other hand, examines linear operators in terms of their interplay. This often gives rise to sets of matrices as just described, and many concepts and questions can thus be studied from both perspectives. This project will first develop new theory at the intersection of the two above-mentioned branches of mathematics. This will lead to a better transfer of methods and results from one area to the other, and a better awareness of the connection between them. The main focus is on operator systems, which are central concepts in both fields. Also a generalization of this notion is an important goal of the project. This will allow to apply several of the known methods and results to questions which do not fit into the framework so far. In a second step we will apply the results to questions from convex algebraic geometry and quantum information theory. We will for example examine the realizability of convex sets as intersection of certain standard convex cones. Such questions are of crucial importance for the geometry of optimization, for example in linear and semidefinite programming. We will further study entanglement of distillation and the PPT2 conjecture within our new framework. Both are important open problems in quantum information theory, which are currently studied with great effort in the scientific community. The combination of semialgebraic geometry and operator algebra is a novel approach to attack the described problems, first promising results have already been achieved recently. This project will help to better develop this approach and unfold its full strength.