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: 2013
• 资助国家: 加拿大
• 起止时间: 2013-01-01 至 2014-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=524515
• 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.

由于M. Gelfand在1941年发表了关于规范环的先驱工作，Banach代数理论已成为功能分析的主要领域。该理论在分析和代数之间存在于其性质上，对现代数学产生了深远的影响。拟议的研究将重点介绍Banach代数的拓扑结构和代数结构，其第二个双重及其理想。我们还将研究半群的休息性和固定点特性。 BANACH代数研究的研究始于1970年代。在这一主题上取得的结果通常代表了Banach代数理论的发展。同种学研究了共同体和共体空间的差异。从传统上讲，串联和共生被视为线性空间，因此，同一个学组只是一个代数对象。但是，这些空间可能自然配备了各种拓扑。人们可以考虑Banach代数的“拓扑”共同体。这是我将为该程序探索的方向。最近引入的各种类型的Banach代数的普遍性舒适性可以解释为第一个共同体学组的“拓扑琐事”，相对于特定拓扑。关于广义的不合适性有许多开放性问题。我将针对它们。 Banach代数的第二个双重双重的Arens产品是Banach代数中的重要对象。 Arens的规律性，拓扑中心和乘数将用于重要的Banach代数。将研究谐波分析中关键对象/概念的Banach代数中的对应物。还将研究各种类型的理想身份。该半群的不变属性，例如WAP（S），LUC（S）和C（S）的不变式均值。 对于半题学半群的固定性特性与固定点特性之间存在着深厚的关系。我将研究这些关系。在Banach空间的封闭子集上的非强度自我映射的半群将特别关注。这种类型的映射的固定点特性在非线性分析领域非常重要。