Theory and applications of operator systems and operator algebras

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The proposed research is in the area of pure mathematics and aims to further develop the theoretical basis for our understanding of certain algebraic structures, particularly C*-algebras and operator systems, that arise in both pure mathematics and theoretical physics. Specifically, this research aims to contribute new information about finite-dimensional operator systems associated with finitely generated groups and to use this information to shed light on a property of C*-algebras known as the weak expectation property. There is potential for applications of this work in the field of quantum information, which is an interdisciplinary research area involving computing science, mathematics, and physics. One way to analyse an abstract mathematical structure is to represent the structure in a different guise. Ideally, if one represents the abstract structure under study in some familiar manner or context, then one can use previous knowledge about well-understood objects to deduce properties or information about the abstract structure under study. The first goal of the proposed research is aimed at doing precisely that. In technical terms, I will be seeking criteria upon which one can decide if a given operator system is a quotient of a matrix operator system. In this scenario the matrix operator system is the familiar object through which one analyses a quotient operator system. In a related vein, the proposed research will examine operator systems arising from finitely generated discrete groups and consider properties of the groups that allow the operator system or generated C*-algebra to detect the weak expectation property. The importance of the proposed research lies, first of all, in the fact that the mathematical structures to be studied have attracted the attention of mathematicians and physicists for nearly a century. Indeed, the field of operator algebras first arose in the early twentieth century out of a need to provide the appropriate mathematical tools for the emerging field of quantum physics. Today, there is a significant amount of international research devoted to operator algebra theory and functional analysis, and the field of quantum information theory has grown enormously in the last decade. The results of the proposed research will be of interest and relevance to such scientists and mathematicians, and the research work itself will provide a vehicle through which mathematical expertise can be developed and retained within Canada.