The Structure of C*-Algebras of Product Systems

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，并将这些结构的离散性质转化为相关算子代数的解析性质。其动机是双重的。一方面，它可以用来测试领域内的猜想：寻找具有某些性质的算子代数归结为产生更简单的动力系统。另一方面，可能会出现跨学科的联系。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空间构造的量子化，类似于量子力学中所做的。在过去的20年里，该社区广泛地研究了与一个离散方向上的动态演化有关的C*-代数。所得到的类包含来自在C*-代数理论中起重要作用的图和动力系统的已知结构。Katsura的工作非常有影响力，允许进一步的发展。让我们简要地描述几个结果，以强调一变量规范不变唯一性定理(GIUT)I.Katsura得到了Cuntz-Pimsner代数有核的充要条件。通过仔细的分析，他还给出了一个计算其K理论的6项精确序列。这种联系正是通过定义协变表示的理想解来提供的。加尾技巧的目的是将非内射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个尾巴来去除图的下沉。对于C*-对应，它首先由Muhly-Tomforde建立，后来由Kakariadis-Katsoulis推广。只要稍加注意，就可以选择尾部来保留子类。三、森田理论有效地匹配了表象，Rieffel引入了一个C*-替代。Blecher-Muhly-Paulsen和Eleftherakis对非自伴代数也发展了类似的理论。Muhly-Solel将其引入到C*-对应的上下文中。利用GIUT，Muhly-Solel和Eleftherakis-Kakariadis-Katsoulis证明了Morita等价是由相关代数继承的。四、Muhly-PaskTomforde从符号动力学出发，将强移位等价提升为正则C*-对应。通过对二部膨胀的GIUT的使用，他们证明了对于Cuntz-Pimsner代数，强移位等价意味着Morita等价，但对于Toeplitz-Pimsner代数和张量代数，却不是这样。这一理论已被Kakariadis-Katsoulis推广到平移等价。一个关键点是内射C*-对应到本质双模的极小扩张。这是通过与内射C*-Dynamics.v类似的直接极限过程建立的。作为共泛对象，C*-包络通常与Cuntz-型代数重合。这是张量代数的情况，正如Katsoulis-Kribs所示，但这并不是排他性的。Kakariadis-Shalit证明了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。几年来，对于更奇异的动力学的协变关系的理解一直是一个谜。然而，最近的发展使我们能够解锁产生严格边界商的正确协变关系。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比及其广泛的应用。我们的研究将在一般水平上进行，但也将在有限秩积系统的水平上进行。

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.