Banach algebra homomorphisms and applications

目前的提议是在抽象调和分析领域，这是一个与局部紧群及其相关的空间和代数的研究有关的数学领域。

The present proposal is in the field of abstract harmonic analysis, a mathematical area concerned with the study of locally compact groups and of the spaces and algebras associated with them. A fundamental question from the early beginnings of the field is to what extent the algebra structure of a given algebra associated to a locally compact group G determines the underlying group G. It is known that the Banach algebra structure completely reflects the structure of G for a series of related algebras such as the group algebra, the measure algebra, as well as their generalizations the Fourier algebra and respectivelly the Fourier-Stieltjes algebra. This is in the sense that two locally compact groups having isometrically isomorphic Banach algebras are necessarily topologically isomorphic. However there are still many open problems if we move away from isometric isomorphisms. We propose to take this line of research in several directions such as investigating the algebra homomorphisms in the new setting of the Figa-Talamanca-Herz algebras and exploring various applications of the description of the completely algebra homomorphisms of the Fourier algebra in terms of piecewise affine maps.