Examples of subhomogeneous Banach and operator algebras

次齐次 Banach 和算子代数的示例

• 批准号: 1943819
• 金额: --
• 依托单位: Lancaster University
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2017
• 资助国家: 英国
• 起止时间: 2017 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-1943819
• 关键词: Examples; subhomogeneous; Banach; operator; algebras

PROJECT DESCRIPTION FOR BLAKE GREEN.[Intended classifications: 70% Mathematical Analysis, 30% Algebra and Geometry]Banach algebras are abstract models that can, in certain settings, provide a unified framework for a range of related problems or phenomena in mathematical analysis. Studying them allows the transfer of ideas and techniques between algebra and analysis.A Banach algebra is said to be n-subhomogeneous if all its irreducible representations have degree at most n. When n=1 this implies the algebra is commutative, and hence many tools from classical commutative algebra and the analytical variants developed by Gelfand, Shilov and others in the 1940s-1950s can be applied. In contrast, for n=2 these tools are inadequate, since we enter the noncommutative realm.The best-understood class of noncommutative Banach algebras is the class of C*-algebras, and n-subhomogeneous C*-algebras admit a very satisfactory theory of their own. However, the foundational results for such objects rely heavily on features that seem to be unique to the C*-setting, and so there has been relatively little attention paid to more general n-subhomogeneous Banach algebras. The need for a better understanding of these more general objects is higlighted by recent work of Choi-Farah-Ozawa (2014), in which the main example is a 2-subhomogeneous operator algebra with certain "exotic" or "pathological" properties.This PhD project lies in the intersection of mathematical analysis with noncommutative ring theory. It has two main objectives, which will be pursued as parallel strands.1) Review and extend some of the existing structural results for subhomogeneous C*-algebras in the more general setting of subhomogeneous operator algebras, thereby putting the Choi-Farah-Ozawa example in proper context. There is currently very little in the existing literature, so the existing methodology of the C*-algebraic setting will need to be refined or augmented with new techniques developed during the PhD. This strand aims to organize some existing folklore but also to map out new territory by means of examples and counterexamples that do not occur in the C*-algebraic setting.2) Explore particular families of subhomogeneous Banach algebras that arise naturally in mathematical analysis, specifically function theory and abstract harmonic analysis: e.g. Banach algebras of matrix-valued differentiable functions, or Banach convolution algebras associated to crystallographic groups. In particular, to what extent can the techniques developed by Gelfand et al. for the n=1 setting be adapted to analyse the structure of these examples? Here, the plan is to start developing new methodology that is a hybrid of Gelfand's functional-analytical techniques with ideas from the world of pure algebra (so-called "P.I. rings").Taken together, the aim of the two strands is to obtain an improved understanding of a natural class of mathematical objects, with both concrete examples and general theory that could be built upon by subsequent researchers.

布莱克·格林的项目描述。【预期分类：70%数学分析，30%代数和几何】Banach代数是抽象模型，在某些情况下，可以为数学分析中的一系列相关问题或现象提供统一的框架。学习它们可以在代数和分析之间传递思想和技术。如果一个Banach代数的所有不可约表示的次数最多为n，则称其为n次齐次代数。当n=1时，这意味着该代数是可交换的，因此经典交换代数中的许多工具以及Gelfand、Shilov等人在20世纪40年代至50年代开发的解析变体都可以应用。相反，对于n=2，这些工具是不够的，因为我们进入了非交换领域。最容易理解的一类非交换Banach代数是C*-代数，并且n次齐次C*-代数承认它们自己的一个非常令人满意的理论。然而，这些对象的基本结果严重依赖于C*设置的独特特征，因此对更一般的n次齐次Banach代数的关注相对较少。Choi-Farah-Ozawa（2014）最近的工作强调了更好地理解这些更一般对象的必要性，其中主要的例子是具有某些“奇异”或“病态”性质的2-亚齐次算子代数。这个博士项目是数学分析与非交换环理论的交叉。它有两个主要目标，这两个目标将作为并行的部分来追求。1)在更一般的次齐次算子代数的背景下，回顾和扩展了一些关于次齐次C*-代数的现有结构结果，从而将Choi-Farah-Ozawa的例子置于适当的背景下。目前在现有的文献中很少，因此C*-代数设置的现有方法将需要在博士期间开发的新技术进行改进或增强。这一链旨在组织一些现有的民间传说，但也通过在C*-代数设置中没有发生的例子和反例子来绘制新的领域。2)探索在数学分析中自然产生的亚齐次巴拿赫代数的特定族，特别是函数理论和抽象调和分析：例如矩阵值可微函数的巴拿赫代数，或与晶体群相关的巴拿赫卷积代数。特别是，Gelfand等人为n=1设置开发的技术在多大程度上可以适用于分析这些示例的结构？在这里，计划是开始开发新的方法，它是Gelfand的函数分析技术与纯代数世界的思想（所谓的“pi环”）的混合。总之，这两方面的目标是通过具体的例子和一般的理论来提高对自然数学对象的理解，从而为后续的研究人员提供基础。