Homomorphisms, representations and amenability of group and Fourier algebras on locally compact groups

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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. With any locally compact group, one can associate several useful collections of mappings into the set of real, or complex, numbers. An operator may be interpreted as a finite or infinite array of numbers, and there are also many relevant collections of operators associated to these locally compact groups. 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. The aforementioned collections of functions and collections of operators are examples of Banach algebras, and this proposal is concerned with the study of various properties of these Banach algebras in relation to properties of their underlying locally compact groups.