Algebras of abstract harmonic analysis

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The mathematical concept of a group is the most basic expression of symmetry. An object like a square admits only finitely many symmetries, whereas a sphere or plane admits infinitely many. In order to keep the analysis of such a group of symmetries more tractable, and to remember more of the spatial features of the underlying sphere or plane, say, one needs to deal with the spatial properties of the symmetry group, a concept we call topology. Hence we obtain topological groups. Furthermore, most natural examples, such as those above, suggest a certain finiteness near individual points, a condition we call local compactness. *******For these locally compact groups it is important to understand their representations, i.e. how they act on other sets with topology. The use of Banach algebras is a tool to encapsulate and help to understand these actions. Such algebras, which include group and measure algebras, weakly almost periodic functions, Fourier and Fourier-Steiltjes algebras, and associated Segal and Beurling algebras all play interesting and distinctive roles in this analysis. My research focusses on the interplay of the structures of locally compact groups and structures of the associated Banach algebras. My most basic question: given a property of a class of locally compact groups, how is that property witnessed in the associated Banach algebras, and vice versa. The resolutions of these questions often force us to use inobvious structures on these Banach algebras, such as operator space structures. For example, amenability,  a fundamental averaging property on groups, is witnessed only in Fourier algebras with an operator space averaging property, and not by a property which can be seen only with Banach algebra techniques.*******Some symmetries observable to physicists require much more subtle symmetries. Remarkably, the mathematical formulation of these, known as quantum groups, also encapsulates generalized Pontryagin duality, which is a framework for understanding the group algebra and Fourier algebra in relation to one another. Hence this more general framework also catches some of my research attention.**