Banach algebra homomorphisms and applications

目前的建议是在抽象调和分析领域，这是一个涉及研究局部紧群和与之相关的空间和代数的数学领域。 从该领域的早期开始，一个基本问题就是与局部紧群G相关的给定代数的代数结构在多大程度上决定了基础群G。众所周知，Banach代数结构完全反映了一系列相关代数的G的结构，例如群代数、测度代数，以及它们在傅立叶代数和傅立叶-斯蒂尔杰斯代数上的推广。这是因为具有等距同构的Banach代数的两个局部紧群必然是拓扑同构的。然而，如果我们离开等距同构，仍然有许多有待解决的问题。 我们建议在几个方向上进行这方面的研究，如研究Figa-Talamanca-Herz代数的新背景下的代数同态，探索用分段仿射来刻画傅里叶代数的完全代数同态的各种应用。

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.