Cohomology in Banach algebras and amenability properties of semigroups

自从M. Gelfand的先驱工作，赋范环发表于1941年，Banach代数理论已成为一个主要领域的功能分析。该理论，站在分析和代数之间的性质，对现代数学产生了深刻的影响。本论文的主要研究内容是Banach代数的拓扑结构、代数结构、第二类Banach代数及其理想。我们还将研究半群的顺从性和不动点性质。Banach代数上同调的研究始于20世纪70年代。在这一问题上取得的成果往往代表着Banach代数理论发展的重大进展。上同调研究上边界空间和上循环空间在上同调群方面的差异。在传统上，上边界和上圈被视为线性空间，因此上同调群只是一个代数对象。然而，这些空间可以自然地配备有各种拓扑结构。可以考虑Banach代数的“拓扑”上同调。这是我将为该计划探索的方向。最近提出的Banach代数的各种类型的广义顺从性可以解释为第一上同调群关于特定拓扑的“拓扑平凡性”。关于广义顺从性，有许多悬而未决的问题。我会瞄准他们。Banach代数的第二对偶上的Arens积是Banach代数中的重要对象。阿伦斯正则性，拓扑中心和乘子将被调查的重要Banach代数。将研究调和分析中关键对象/概念在Banach代数中的对应物。各种类型的近似身份的理想也将研究。该程序将研究半群S的顺从性，如WAP（S），LUC（S）和C（S）上的不变平均。 拓扑半群的顺从性与不动点性质之间有着深刻的联系。我将研究这些关系。Banach空间的闭子集上的非扩张自映射半群将特别受到关注。这类映射的不动点性质在非线性分析领域中是极其重要的。

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.