Banach algebras in abstract harmonic analysis

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This proposal lies in the area of abstract harmonic analysis, which is largely the study of locally compact (topological) groups and their associated Banach algebras. Locally compact groups are abstract mathematical objects, which may be associated with the symmetries of a geometric figure, or the rigid motion of objects in nature. They have many applications, both to the mathematical and physical sciences. A Banach algebra is a collection of elements such that each individual element has a length, and such that any two elements can be added or multiplied. These Banach algebras are often comprised of mappings into the set of real, or complex, numbers, and they are also often comprised of operators, which may be viewed as finite, or infinite, arrays of numbers. There are many interesting and useful Banach algebras naturally associated with any locally compact group, and a general theme in this area of research is to study the relationship between various properties of locally compact groups and their associated Banach algebras. In many ways, this theme is prominent in this research program. One purpose of mathematics is to find simple descriptions of complex phenomena. In particular, the search for simple descriptions of (complicated) structure-preserving maps between similar mathematical objects is a fundamental objective in virtually every area of mathematics. For many years, abstract harmonic analysts have searched for attractive descriptions of the structure-preserving maps between various Banach algebras built over locally compact groups, and this is one of the purposes of this research program. In recent years, modern tools have generated significant progress with this endeavour, and this proposal includes new approaches to solving some of these challenging problems.