Groupoid models for diagrams of groupoid correspondences and their C*-algebras

用于群群对应图及其 C* 代数的群群模型

• 批准号: 529300231
• 负责人: Professor Dr. Ralf Meyer
• 金额: --
• 依托单位: Mathematisches Institut (MI)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 资助国家: 德国
• 项目状态: 未结题
• 来源: https://gepris.dfg.de/gepris/projekt/529300231?language=en
• 关键词: Groupoid; models; diagrams; groupoid; correspondences

Diagrams of groupoid correspondences describe a very general kind of dynamical systems, which contains self-similarities of groups and topological graphs as special cases. Self-Similar groups are important invariants of dynamical systems because they are purely algebraic and still may determine all of the dynamics on a Julia set. Graph C*-algebras are an important class of C*-algebras because many of their properties may be read off the underlying graphs. In order to handle more C*-algebras in similar ways, several generalisations of graph C*-algebras have been introduced in recent years. Diagrams of groupoid correspondences allow to treat all these classes of examples in a uniform way and also provide more examples that are similarly amenable to study, that is, they inherit many important properties of graph C*-algebras. Just as a group action may be encoded in a groupoid, which then gives rise to a C*-algebra, so a diagram of groupoid correspondences may be described in one groupoid, called its groupoid model. The existing constructions of this groupoid, however, only work under an extra assumption. A suitable generalisation to all diagrams is an important goal of this project. Alternatively, one may first translate a diagram of groupoid correspondences to the realm of C*-algebras or plain algebras and then map the diagram to a single C*-algebra or algebra, called its a covariance algebra. How is this covariance algebra related to the groupoid C*-algebra or Steinberg algebra of the groupoid model? This is the second central question in this proposal. It is known that these objects agree under some assumptions, but not always. Finally, we will investigate some important properties of groupoid models, which allow to decide, for instance, whether the groupoid model is simple, that is, its only quotients are the obvious ones.

广群对应图描述了一类非常普遍的动力系统，它包含了群和拓扑图的自相似性作为特例。 自相似群是动力系统的重要不变量，因为它们是纯代数的，并且仍然可以确定Julia集上的所有动力学。 图C ~*-代数是C ~*-代数的一个重要类别，因为它们的许多性质可以从底层图中读出。 为了以类似的方式处理更多的C*-代数，近年来引入了图C*-代数的几种推广。 广群对应图允许以统一的方式处理所有这些类的例子，并且还提供了更多类似的适合研究的例子，也就是说，它们继承了图C*-代数的许多重要性质。正如群作用可以被编码在群胚中，然后产生C*-代数，所以群胚对应的图可以在一个群胚中描述，称为它的群胚模型。 然而，这个广群的现有构造只在一个额外的假设下工作。 一个合适的概括到所有的图表是这个项目的一个重要目标。 或者，我们可以先将广群对应的图转换到C*-代数或普通代数的领域，然后将图映射到单个C*-代数或代数，称为协方差代数。 这个协方差代数与广群模型的广群C*-代数或Steinberg代数有什么关系？ 这是本提案的第二个核心问题。 已知这些对象在某些假设下一致，但不总是一致。最后，我们将研究群胚模型的一些重要性质，例如，这些性质可以决定群胚模型是否简单，也就是说，它的唯一等价物是明显的等价物。