Theory and applications of operator systems and operator algebras

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The focus of my research program concerns the part of functional analysis that encompasses the theory of bounded linear operators acting on Hilbert spaces, C*-algebras and von Neumann algebras, and abstract operator systems. These particular fields are very active areas of research that engage a significant number of high-quality mathematicians worldwide. Indeed, these areas also represent branches of mathematics in which Canada is known internationally to be particularly strong. The long-term objectives of the present research proposal are to make advances in operator algebra theory overall and, through those advances, to make novel, useful contributions of the theory of operator systems and operator algebras to related fields such as single-operator theory, quantum information theory, free convexity (also known as operator convexity, matrix convexity, and noncommutative convexity), and noncommutative function theory. As part of this program, I aim to strengthen the interactions between these seemingly diverse subfields, and to engage highly qualified personnel in those aspects of this research that have the potential to develop a knowledge base that has sustained value beyond the scope of the presently proposed work. In the near term, I am proposing to continue the trajectory of the research undertaken during my current NSERC Discovery Grant, with a particular emphasis on achieving the following objectives: (i) understanding the nature of the matrix-state spaces of operator system tensor products; (ii) advancing free topology, and determining a framework for free measure theory and integral representations of operator-convex functions on free convex sets; (iii) the analysis of the spectra of quantum channels on the fermion algebra; (iv) developing new measures of quantum noise. Expected outcomes of this research include: (a) the training of new personnel equipped with a powerful skill set in mathematics, with a particular emphasis on developing the mathematical talents of women, minority peoples, or people with disabilities; and (b) the advancement of knowledge in a branch of mathematics that is studied and applied worldwide by mathematicians, theoretical physicists, and computer scientists.