Studies in Operator Algebras

算子代数研究

• 批准号: 0401043
• 负责人: Roger Smith
• 金额: $ 15万
• 依托单位: Texas A&M Research Foundation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2008-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0401043&HistoricalAwards=false
• 关键词: Studies; Operator; Algebras

The proposer will continue his study of the longstanding Kadison--Ringroseconjecture on the cohomology of von Neumann algebras. He will use and refine some recently developed techniques needed to show the complete boundedness of certain multilinear operators. Investigations of the cohomologygroups have led naturally to the recently introduced concept of norming subalgebras, and this topic will be pursued since it applies directly to both cohomology and the bounded projection problem. In a different, but related, direction, maximal abelian subalgebras (masas) and general subalgebras of von Neumann algebras will be studied. In joint work, the proposer has introduced the concept of strongly singular masas, and has shown that many finite factors arising from hyperbolic groups (a class which includes the free groups) have such masas. The results so far obtained indicate that the theory can now expand to study general subalgebras. This relates directly to the structure of factors (the building blocks of operator algebras), and the overall goal of these proposed areas of research is to increase our understanding in this area. Building on previously accomplished work, problems in the theory of topological entropy for automorphisms will be studied. Automorphisms are the most basic objects associated to operator algebras, and they can reveal different facets of the same underlying object. The entropy is a numerical constant that distinguishes different automorphisms.Previous joint work of the proposer showed that the current theory, based on completely positive maps, can be better reformulated in terms of complete contractions. This puts the theory into a much more flexible situation where more general and powerful tools can be brought to bear on the problems of the field, notably relating to crossed products by automorphism groups. 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 web. 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, and here topological entropy plays an important role. The project is mainly concerned with the theoretical underpinnings of operator algebras, but the proposed work in these areas could impact some of these more concrete areas, since the finite factors are those operator algebras which most closely model matrices.

该提议将继续他的长期研究卡迪森-环罗第二猜想在冯·诺伊曼代数的上同调。他将使用和改进一些最近发展的技术来证明某些多线性算子的完全有界性。对上同调群的研究自然导致了最近引入的赋范子代数的概念，这个主题将继续进行，因为它直接适用于上同调和有界投影问题。在一个不同的，但相关的方向，极大阿贝尔子代数（masas）和冯·诺伊曼代数的一般子代数将被研究。在联合工作中，提出了强奇异masas的概念，并证明了由双曲群（包括自由群的一类）产生的许多有限因子具有强奇异masas。目前得到的结果表明，该理论现在可以扩展到研究一般子代数。这直接关系到因子的结构（算子代数的构建块），这些拟议研究领域的总体目标是增加我们对这一领域的理解。在先前已完成的工作的基础上，将研究自同构的拓扑熵理论中的问题。自同构是与算子代数相关的最基本对象，它们可以揭示同一底层对象的不同方面。熵是区分不同自同构的数值常数。先前的联合工作表明，目前的理论，基于完全正映射，可以更好地重新表述为完全收缩。这使得该理论处于一种更加灵活的情况，在这种情况下，可以使用更通用和更强大的工具来处理该领域的问题，特别是与自同构群的交叉产物有关的问题。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网络搜索引擎都有应用。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的矩阵，称为线性算子，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，拓扑熵在其中起着重要的作用。该项目主要关注算子代数的理论基础，但在这些领域提出的工作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矩阵模型的算子代数。