Grant: Discovery Grants Program - Individual

Operator theory and the theory of operator algebras arose in the twentieth century, in part, out of a need to provide the mathematical tools for the newly emerging field of quantum physics. While there remain important connections between this branch of mathematics and physics, the quantisation of physics has carried over to mathematics, where certain mathematical structures are revealed (or classified) only by taking into account their features as ""noncommutative"" entities. This view is especially prevalent in modern analysis and algebra, and in operator algebra theory in particular. The goal of my research program is to advance operator algebra theory through the study of order, convexity, and topology, and to apply operator-algebraic methods to problems in single operator theory. It is very common to study a C*-algebra A or an operator system E by way of one of its enveloping structures. My research program has been investigating such structures that arise from the so-called Hamana theory of injective envelopes. Essentially the Hamana theory embeds a C*-algebra A or an operator system E into an injective C*-algebra which is, in a sense that can be made precise, the smallest injective C*-algebra that contains A or E. The process of passing from A to its injective envelope I(A) is very much like the passage from the rational numbers to the real numbers. Of course the noncommutativity of C*-algebras adds a level of complexity that is not seen with ordinary number systems. The enveloping structures I am considering are local multiplier algebras, regular monotone completions, C*-envelopes, and injective envelopes. Each of these structures make useful appearances in operator algebra theory and operator theory in different ways. However, none of these algebras are well understood. My research consists of a number of avenues of investigation that will, I believe, reveal some of the fundamental properties of these enveloping algebras.