Cohomology in Banach algebras and amenability properties of semigroups

自从1941年M.Gelfand关于赋范环的开创性工作发表以来，Banach代数理论已经成为泛函分析的一个主要领域。这一理论本质上介于分析和代数之间，对现代数学产生了深远的影响。我们的研究将集中在Banach代数的拓扑结构和代数结构、它们的第二对偶以及它们的理想。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。在这个问题上取得的成果往往代表着Banach代数理论发展的重大进步。上同调研究上同调群中上边界空间和上循环空间的差异。在传统上，上边界和上循环被视为仅仅是线性空间，因此上同调群仅仅是一个代数对象。然而，这些空间可能自然而然地配备了各种拓扑。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”。关于广义的顺从性，有许多未决的问题。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。我们将研究重要的Banach代数的Arens正则性、拓扑中心和乘子。我们将研究调和分析中关键对象/概念的Banach代数中的对应物。还将研究各种类型的理想的近似恒等式。本程序将研究半群S的顺从性性质，如在WAP(S)、LUC(S)和C(S)上的不变平均。半拓扑半群的顺从性与不动点性质之间有着深刻的联系。我会研究这些关系。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非常重要的。

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.