Banach algebra homomorphisms and applications

目前的建议是在该领域的抽象谐波分析，数学领域关注的研究局部紧群和空间和代数与他们有关。 从域的早期开始的一个基本问题是，与局部紧群G相关联的给定代数的代数结构在多大程度上决定了基础群G。Banach代数的结构完全反映了群代数、测度代数及其推广的Fourier代数和Fourier-Stieltjes代数等一系列相关代数G的结构。这是在这个意义上说，两个局部紧群具有等距同构的Banach代数必然拓扑同构。然而，如果我们离开等距同构，仍然有许多开放的问题。 我们建议采取这条线的研究在几个方向，如调查的代数同态在新的设置的Figa-Talamanca-Herz代数和探索各种应用程序的描述完全代数同态的傅立叶代数分段仿射映射。

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.