Cohomology in Banach algebras and amenability properties of semigroups

自从1941年Gelfand关于赋范环的开创性工作发表以来，巴拿赫代数理论已经成为泛函分析的一个主要领域。这个理论在本质上介于分析和代数之间，对现代数学产生了深刻的影响。本文主要研究了Banach代数的拓扑结构、代数结构、第二对偶及其理想。我们还研究了半群的可适应性质和不动点性质。Banach代数上同调的研究始于20世纪70年代。在这个问题上取得的成果往往代表了巴拿赫代数理论发展的重大进展。上同调研究共边空间和共环空间在上同调群上的差异。在传统上，共边和共环被视为线性空间，因此上同群只是一个代数对象。然而，这些空间可以自然地配备各种拓扑结构。我们可以考虑Banach代数的“拓扑”上同调。这是我将为这个项目探索的方向。最近引入的各种类型的Banach代数的广义适应性可以解释为第一上同群相对于特定拓扑的“拓扑平凡性”。关于广义适从性有许多悬而未决的问题。我会瞄准他们。巴拿赫代数第二对偶上的阿仑积是巴拿赫代数中的重要对象。本文将研究重要的Banach代数的Arens正则性、拓扑中心和乘子。在调和分析中的关键对象/概念的对应物巴拿赫代数将被研究。各种类型的理想近似恒等式也将被研究。本课程将研究半群S在WAP(S)、LUC(S)和C(S)上的不变量均值等适应性性质。半拓扑半群的易动性和不动点性之间存在着深刻的联系。我将研究这些关系。Banach空间的闭子集上的非扩张自映射的半群将被特别关注。这类映射的不动点性质在非线性分析领域中具有极其重要的意义。

Since M. Gelfand's pioneer work on normed rings was published in 1941, Banach algebra theory has become a major field in functional analysis. The theory, standing between analysis and algebra in its nature, has had a deep influence upon modern mathematics. The proposed research will focus on topological and algebraic structures of Banach algebras, their second duals and their ideals. We will also investigate the amenability properties and fixed point properties of semigroups. The study of cohomology in Banach algebras began in 1970s. Results achieved on this subject often represent significant progress in the development of Banach algebra theory. Cohomology studies the difference, in terms of cohomology groups, of the spaces of coboundaries and cocycles. In tradition, coboundaries and cocycles are treated as just linear spaces, and hence a cohomology group is simply an algebraic object. However, these spaces may be naturally equipped with various topologies. One can consider "topological" cohomology for Banach algebras. This is the direction in which I will explore for the program. The recently introduced various types of generalized amenability for Banach algebras may be interpreted as "topological triviality" of the first cohomology group with respect to specific topologies. There are many open problems regarding generalized amenability. I will target them. Arens products on the second dual of a Banach algebra are significant objects in Banach algebras. Arens regularity, topological centers and multipliers will be investigated for important Banach algebras. Counterparts in Banach algebras of crucial objects/notions in harmonic analysis will be investigated. Various types of approximate identities for ideals will also be studied. Amenability properties of a semigroup S, such as invariant means on WAP(S), LUC(S) and C(S), will be studied in the program. There are deep relations between amenability properties and fixed point properties for a semitopological semigroup. I will study these relations. Semigroups of non-expansive self mappings on a closed subset of a Banach space will be particularly concerned. Fixed point property of this type of mappings is extremely important in the field of nonlinear analysis.