The Structure of C*-Algebras of Product Systems

乘积系统的 C* 代数的结构

• 批准号: 2441268
• 金额: --
• 依托单位: Newcastle University
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2441268
• 关键词: Structure; Algebras; Product; Systems

A major trend in the area of operator theory is the use of operator algebras for quantizing geometrical structures and translating discrete properties of the structures into analytic properties of the related operator algebras. The motivation is two-fold. On the one hand, it can be used to test conjectures within the field: finding operator algebras with certain properties reduces to producing simpler dynamical systems. On the other hand, interdisciplinary links can emerge. For example, classification problems can be tested through isomorphisms of algebras. To this end, the quantizations must reflect rigidly the geometrical behaviour of the structure.In our study we consider a quantization via a Fock space construction, similar to what is done in quantum mechanics. In the past 20 years, the community has extensively studied C*-algebras that relate to dynamics evolving in one discrete direction. The obtained class contains previously known constructions arising from graphs and dynamical systems that play a prominent role in the theory of C*-algebras. Katsura's work has been very influential, allowing for further developments. Let us give a short description of several results to emphasize on the impact of the one-variable Gauge Invariant Uniqueness Theorem (GIUT).i. Katsura achieved necessary and sufficient conditions for nuclearity of Cuntz-Pimsner algebras. Through a careful analysis, he also produced a 6-term exact sequence for computing its K-theory. The link is provided exactly through the solutions of ideals that define the covariant representations.ii. The purpose of the tail-adding technique is to dilate a non-injective C*-correspondence to an injective one so that their Cuntz-Pimsner algebras are Morita equivalent. This is in analogy to removing sinks of graphs by adding infinite tails. For C*-correspondences, it was first established by Muhly-Tomforde and later extended by Kakariadis-Katsoulis. With some care, the tail can be chosen to preserve sub-classes. iii. Morita theory effectively matches representations, and a C*-alternative has been introduced by Rieffel. A similar theory has been developed for non-selfadjoint algebras by Blecher-Muhly-Paulsen and Eleftherakis. Muhly-Solel imported it to the context of C*-correspondences. By using the GIUT it was shown by Muhly-Solel and Eleftherakis-Kakariadis-Katsoulis that Morita equivalence is inherited by the related algebras. iv. Taking motivation from symbolic dynamics, Muhly-Pask-Tomforde lift the strong shift equivalence to regular C*-correspondences. By using the GIUT on the bipartite inflation, they showed that strong shift equivalence implies Morita equivalence for the Cuntz-Pimsner algebras, but not for the Toeplitz-Pimsner algebras and tensor algebras. This theory has been extended to shift equivalence by Kakariadis-Katsoulis. A key point is the minimal extension of an injective C*-correspondence to an essential bimodule. This was established through a direct limit process similar to injective C*-dynamics.v. Being a co-universal object, the C*-envelope often coincides with a Cuntz-type algebra. This is the case for the tensor algebras, as shown by Katsoulis-Kribs, but this is not exclusive. As shown by Kakariadis-Shalit, the C*-envelope of the tensor algebra of a factorial language is the quotient by generalized compacts. The use of the GIUT is crucial to achieve these results.The understanding of the covariant relations for more exotic dynamics had remained a mystery for several years. However, recent developments have allowed us to unlock the correct covariant relations that produce rigid boundary quotients. Our motivation in this project is to explore the multi-variable analogues of the Gauge Invariant Uniqueness Theorem and its vast applications. Our research will be carried at the general level but also at the level of product systems of finite rank.

算子理论领域的一个主要趋势是使用算子代数来量化几何结构，并将结构的离散性质转化为相关算子代数的分析性质。动机是双重的。一方面，它可以用来测试领域内的代数：寻找具有某些性质的算子代数可以简化为产生更简单的动力系统。另一方面，跨学科的联系可以出现。例如，分类问题可以通过代数的同构来测试。为此，量子化必须严格反映结构的几何行为。在我们的研究中，我们考虑通过Fock空间结构的量子化，类似于量子力学中所做的。在过去的20年里，社区已经广泛研究了C*-代数，涉及到一个离散方向上的动态演化。所获得的类包含以前已知的结构所产生的图形和动力系统，在C*-代数理论中发挥了突出的作用。卡茨的工作一直非常有影响力，允许进一步的发展。让我们给出几个结果的简短描述，以强调单变量规范不变唯一性定理（GIUT）的影响。Katerina得到了Cuntz-Pimsner代数有核的充分必要条件。通过仔细的分析，他还产生了一个6项精确序列来计算其K理论。该链接是通过定义协变表示的理想的解决方案。加尾技术的目的是将非内射C*-对应扩张为内射C*-对应，使得它们的Cuntz-Pimsner代数是Morita等价的.这类似于通过添加无限的尾部来移除图的汇。对于C*-对应，它首先由Muhly-Tomforde建立，后来由Kakariadis-Katsoulis扩展。如果小心的话，可以选择尾部来保存子类。三.森田理论有效地匹配表示，和C*-替代已介绍了里费尔。Blecher-Muhly-Paulsen和Eleftherakis也为非自伴代数发展了类似的理论。Muhly-Solel将其导入到C*-对应的上下文中。Muhly-Solel和Eleftherakis-Kakariadis-Katsoulis利用GIUT证明了Morita等价可由相关代数继承。四. Muhly-Pask-Tomforde从符号动力学出发，将强移位等价提升到正则C*-对应。通过在二部膨胀上使用GIUT，他们证明了强移位等价对于Cuntz-Pimsner代数意味着Morita等价，但对于Toeplitz-Pimsner代数和张量代数不意味着Morita等价。Kakariadis-Katsoulis将这一理论扩展到了移位等价。一个关键点是内射C*-对应到本质双模的最小扩张。这是通过类似于内射C*-动力学的直接极限过程建立的。v.作为一个共泛对象，C*-包络经常与一个Cuntz型代数相吻合。这是张量代数的情况，如Katsoulis-Krribs所示，但这不是唯一的。如Kakariadis-Shalit所示，阶乘语言的张量代数的C*-包络是广义紧的商。GIUT的使用是实现这些结果的关键。对于更奇异动力学的协变关系的理解多年来仍然是一个谜。然而，最近的发展使我们能够解锁正确的协变关系，产生刚性边界约束。我们在这个项目中的动机是探索规范不变唯一性定理的多变量类似物及其广泛的应用。我们的研究将在一般的水平，但也在有限秩的产品系统的水平。