The Structure of C*-Algebras of Product Systems

在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解析性质。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，跨学科的联系也会出现。例如，分类问题可以通过代数的同构进行测试。为此，量化必须严格反映结构的几何特性。在我们的研究中，我们考虑通过Fock空间构造进行量子化，类似于在量子力学中所做的。在过去的20年里，学界对C*-代数进行了广泛的研究，这些代数与一个离散方向的动力学演化有关。所获得的类包含先前已知的由图和动力系统产生的结构，这些结构在C*-代数理论中起着突出的作用。桂的工作非常有影响力，允许进一步的发展。为了强调单变量规范不变唯一性定理（GIUT）的影响，我们将给出几个结果的简短描述。Katsura获得了Cuntz-Pimsner代数核性的充分必要条件。通过仔细的分析，他还提出了一个6项精确序列来计算其k理论。这种联系是通过定义协变表示的理想解精确地提供的。加尾技术的目的是将一个非内射C*对应扩展为一个内射C*对应，使它们的Cuntz-Pimsner代数是Morita等价的。这类似于通过添加无限尾来消除图形的下沉。对于C*对应，它首先由Muhly-Tomforde建立，后来由Kakariadis-Katsoulis推广。小心点，可以选择尾巴来保留子类。3。Morita理论有效匹配表征，Rieffel引入了C*替代。Blecher-Muhly-Paulsen和Eleftherakis已经为非自伴代数发展了一个类似的理论。Muhly-Solel将其导入到C*-通信的上下文中。Muhly-Solel和eleeftherakis - kakariadis - katsoulis利用GIUT证明了相关代数继承了Morita等价。从符号动力学出发，muhly - park - tomforde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Cuntz-Pimsner代数意味着Morita等价，但对Toeplitz-Pimsner代数和张量代数不意味着Morita等价。Kakariadis-Katsoulis将这一理论扩展到平移等价。关键的一点是一个内射C*对应于一个本质双模的最小扩展。这是通过类似于注入C*-动力学的直接极限过程建立的。作为一个共宇宙对象，C*包络常常与康茨型代数相吻合。这是张量代数的情况，如Katsoulis-Kribs所示，但这不是排他的。如Kakariadis-Shalit所示，阶乘语言的张量代数的C*包络是广义紧的商。使用GIUT对于实现这些结果至关重要。对更多奇异动力学的协变关系的理解多年来一直是一个谜。然而，最近的发展使我们能够解开产生刚性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，也将在有限秩产品系统的水平上进行。

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.