Cohomology in Banach algebras and amenability properties of semigroups

Banach代数中的上同调和半群的顺从性

• 批准号: 238949-2011
• 负责人: Zhang, Yong
• 金额: $ 0.73万
• 依托单位: University of Manitoba
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570533
• 关键词: Cohomology; Banach; algebras; amenability; properties

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.

自 1941 年 M. Gelfand 发表关于赋范环的开创性著作以来，Banach 代数理论已成为泛函分析的一个主要领域。该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。拟议的研究将集中于巴拿赫代数的拓扑和代数结构、它们的第二对偶及其理想。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。该主题取得的成果通常代表了巴拿赫代数理论发展的重大进展。上同调研究上同调群、共边界和余循环空间的差异。在传统中，余界和余循环被视为线性空间，因此上同调群只是一个代数对象。然而，这些空间可能自然地配备有各种拓扑。人们可以考虑巴拿赫代数的“拓扑”上同调。这也是我节目要探索的方向。最近引入的各种类型的巴纳赫代数的广义顺应性可以被解释为第一上同调群相对于特定拓扑的“拓扑平凡性”。关于普遍的顺应性存在许多悬而未决的问题。我会瞄准他们。巴拿赫代数的第二对偶上的阿伦斯积是巴拿赫代数中的重要对象。对于重要的巴拿赫代数，将研究阿伦斯正则性、拓扑中心和乘子。将研究调和分析中关键对象/概念的巴拿赫代数的对应部分。还将研究理想的各种类型的近似恒等式。该程序将研究半群 S 的顺应性性质，例如 WAP(S)、LUC(S) 和 C(S) 上的不变均值。 半拓扑半群的顺从性和不动点属性之间存在着深刻的关系。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受到特别关注。此类映射的不动点性质在非线性分析领域极其重要。