Grant: Discovery Grants Program - Individual

My mathematical research is in the closely-connected areas of operator algebras, operator theory, and noncommutative convexity. Operator algebras are mathematical objects which lie at the intersection of linear algebra, mathematical analysis, abstract algebra, and the theory of quantum mechanics and quantum information. Broadly, I study linear operators in finite and infinite dimensions on what is known as a Hilbert space. In scientific applications, a Hilbert space can represent the possible states of a quantum system, or a space of discrete or continuous time-signals, and an operator algebra can represent a collection of quantum observables, or operations on time-signals. My proposed research plan for the next five years can be broken down into three interconnected pillars of investigation. The first concerns non-selfadjoint operator algebras and their C*-covers. The second concerns noncommutative convexity and its relationship to operator system theory and duality. The final pillar is the interface between the first two. An operator algebra is a sub-algebra of the linear operators on the Hilbert space. Much is known about C*-algebras, which are operator algebras that are symmetric in the sense that they are closed under the adjoint operation on operators. In physics, the adjoint can represent time-reversal, or the opposite or dual of an operator. In recent joint work with Christopher Ramsey at MacEwan University, we demonstrated that studying the collection of C*-algebras generated by an operator algebra-called C*-covers, that may not be adjoint-closed is an interesting and productive thing to do, but also that this study has fundamental limitations. In further joint and solo work, I propose further study of these C*-covers, including important subclasses of C*-covers and what collections of C*-covers are possible. Spaces of operators, called operator systems, are algebraic objects equivalent to geometric objects called noncommutative (nc) convex sets. Nc convex sets contain matrices of various sizes. Recently, with Matthew Kennedy and Nicholas Manor at the University of Waterloo, I showed that nc convex sets are a route towards understanding duality between operator systems. I propose further study of duality regarding the important algebraic operation of tensor product using nc convexity. This is a long term plan involving much theory-building, which should further the study of operator systems. Finally, my overarching goal is to unify the study of operator algebras with nc convex geometry. Just as for operator systems and nc convex sets, there must be a corresponding geometric object to operator algebras in noncommutative convexity. Nobody has yet written down this correspondence, so I intend to codify this. This will allow us to transport powerful geometric tools from nc convexity to the study of operator algebras.