Perturbations of Operator Algebras and Related Topics

该项目有两个主要重点领域。第一个问题涉及算子代数的扰动理论，特别是长期存在的Kadison-Kastler问题，该问题询问在适当度量中接近的算子代数是否必须同构（甚至是酉等价）。这是一个更一般和更困难的问题的一部分：当两个代数是相同的（同构的），我们如何才能认识到这是如此？困难的出现是因为同一个代数可以有许多不同的，看似无关的表示。在最近与合著者的联合工作中，首席研究员在Kadison-Kastler问题上取得了相当大的进展，他计划在几个方向继续这项工作：封闭交叉积代数的同构结果，封闭代数的K理论，封闭代数的共享属性，以及封闭包含是否导致嵌入的问题。最近的进展来代数具有相似性的性质，这有一个重新制定的上同调条款。因此，第二个主要的焦点领域将是凯迪森-林格罗斯问题，它询问某些上同调群（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映射和多线性映射的理论，也与有界投影性质有关。矩阵是数字的一般化，它被用来解方程，现在从计算机图形学到互联网搜索引擎都有应用。在量子力学的数学表述中，冯·诺依曼发现他需要被称为线性算子的矩阵的无限维版本，而线性算子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可以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水平上的完全有界和完全正映射的理论，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这些更具体的领域，因为有限因子是那些最接近模型矩阵代数的算子代数。主要研究员工作的一个重要方面是培训博士后研究人员和博士生。由首席研究员指导的大多数年轻人现在都在教职上，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该国的技术未来至关重要，因此首席研究员将继续发挥指导年轻数学家的核心作用。

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.