Perturbations of Operator Algebras and Related Topics

这个项目有两个主要的重点领域。第一个是关于算子代数的微扰理论，特别是长期存在的卡迪逊-卡斯特勒问题，该问题询问在合适度量中接近的算子代数是否必须是同构的（甚至是酉等价的）。这是一个更普遍和更困难的问题的一部分：当两个代数相同（同构）时，我们如何认识到这是如此？困难的出现是因为相同的代数可以有许多不同的，看似不相关的表示。在最近与合著者的合作中，首席研究员在Kadison-Kastler问题上取得了相当大的进展，他计划在几个方向上继续这项工作：紧密交叉积代数的同构结果，紧密代数的k理论，紧密代数的共有性质，以及紧密包含是否导致嵌入的问题。最近的进展是关于具有相似性质的代数，这在上同调项中有一个重新表述。因此，第二个主要关注的领域将是卡迪逊-林罗斯问题，该问题询问某些上同群（作为理想性质的障碍）是否必须消失。这将涉及到完全有界线性和多线性映射的理论，也连接到有界投影性质。算子代数的现代研究有两个主要来源。矩阵是数字的泛化，被用来解方程，现在从计算机图形学到互联网搜索引擎都有应用。在用数学方法表述量子力学时，冯·诺伊曼发现他需要无限维矩阵的线性算子，这在算子代数中得到了最好的研究。此外，量子力学系统的时间演化开始用自同构群的交叉积来表示。新兴的量子计算理论基本上是基于矩阵级的完全有界和完全正映射理论，这些主题是将要进行的大部分工作的基础。因此，在这些领域获得的结果可能会影响一些更具体的领域，因为有限因子是那些最接近模拟矩阵代数的算子代数。首席研究员工作的一个重要方面是培养博士后研究人员和博士生。大多数接受过首席研究员指导的年轻人现在都担任教职，他们正在培训下一代科学家和工程师。一支受过科学和数学训练的劳动力队伍对国家的技术未来至关重要，因此首席研究员将继续在指导年轻数学家方面发挥核心作用。

This project has two main areas of focus. The first of these concerns the perturbation theory of operator algebras with particular reference to the longstanding Kadison-Kastler problem, which asks whether operator algebras that are close in a suitable metric must be isomorphic (or even unitarily equivalent). This is part of a more general and very difficult problem: when two algebras are the same (isomorphic), how can we recognize that this is so? The difficulty arises because the same algebra can have many different, seemingly unrelated, representations. In recent joint work with coauthors, the principal investigator has made considerable progress on the Kadison-Kastler problem, and he plans to continue this work in several directions: an isomorphism result for close crossed product algebras, the K-theory of close algebras, the shared properties of close algebras, and the question of whether close containments lead to embeddings. Recent progress has come for algebras that have the similarity property, and this has a reformulation in cohomological terms. Thus the second main area of focus will be the Kadison-Ringrose problem, which asks whether certain cohomology groups (which act as obstructions to desirable properties) must vanish. This will involve the theory of completely bounded linear and multilinear maps and also connects to the bounded projection property.The modern study of operator algebras has evolved from two main sources. Matrices, which are generalizations of numbers, were introduced to solve equations and now find applications from computer graphics to search engines for the internet. In formulating quantum mechanics mathematically, von Neumann found that he needed infinite-dimensional versions of matrices called linear operators, which were best studied in operator algebras. Moreover, the time-evolution of quantum mechanical systems came to be expressed in terms of the crossed product by groups of automorphisms. The emerging theory of quantum computation is substantially based on the theory of completely bounded and completely positive maps at the matrix level, and these topics underlie much of the work that will be undertaken. Thus, the results obtained in these areas are likely to impact some of these more concrete areas, since the finite factors are those operator algebras that most closely model matrix algebras. An important aspect of the principal investigator's work has been the training of postdoctoral researchers and doctoral students. Most of the young people who have been mentored by the principal investigator are now in faculty positions where they are training the next generation of scientists and engineers. A scientifically and mathematically trained workforce is essential for the technological future of the country, so the principal investigator will continue to give a central role to the mentoring of young mathematicians.