Examples of subhomogeneous Banach and operator algebras

Blake Green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，在某些设置下，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。研究它们可以在代数和分析之间传递思想和技巧。如果一个Banach代数的所有不可约表示至多具有n次n次，则称它是n次齐次的。当n=1时，这意味着该代数是可交换的，因此可以使用经典交换代数和Gelfand，Shilov等人在20世纪40-50年代发展的分析变体的许多工具。相反，对于n=2，这些工具是不够的，因为我们进入了非对易领域。最好理解的非对易Banach代数类是C*-代数类，而n-次齐次C*-代数有一个非常令人满意的理论。然而，这类对象的基本结果在很大程度上依赖于似乎是C*-环境所特有的特征，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，在该工作中，主要的例子是具有某些“奇异”或“病理”性质的2-次齐次算子代数。这个博士项目位于数学分析和非对易环论的交集。它有两个主要目标，这两个目标将作为并行链1)回顾和推广次齐次C*-代数在更一般的次齐次算子代数的设置下的一些已有的结构结果，从而将Choi-Farah-Ozawa的例子放在适当的上下文中。目前在现有的文献中很少有，所以现有的C*-代数设置的方法将需要在博士期间开发的新技术来改进或扩充。这篇文章的目的是组织一些现有的民间传说，但也通过C*-代数环境中不出现的例子和反例来绘制新的领域。2)探索在数学分析中自然产生的特殊的次齐次Banach代数族，特别是函数论和抽象调和分析：例如，矩阵值可微函数的Banach代数，或与晶体群相关的Banach卷积代数。特别是，Gelfand等人开发的技术在多大程度上可以。对于n=1设置，是否适用于分析这些示例的结构？在这里，我们的计划是开始开发一种新的方法论，它是盖尔芬德的函数分析技术与纯代数界的思想(所谓的“P.I.环”)的混合体。综合起来，这两种方法的目标是更好地理解一类自然的数学对象，既有具体的例子，也有可供后续研究人员建立的一般理论。

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.