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.

提议者将继续研究长期以来关于冯诺依曼代数上同调的卡迪森-环玫瑰猜想。他将使用和完善一些最近开发的技术来显示某些多线性算子的完全有界性。对上同调群的研究自然地引出了最近引入的规范子代数的概念，并且该主题将被继续研究，因为它直接适用于上同调和有界投影问题。在不同但相关的方向上，将研究冯诺依曼代数的最大阿贝尔子代数（masas）和一般子代数。在联合工作中，提议者引入了强奇异masas的概念，并证明了双曲群（包括自由群的一类）产生的许多有限因子都具有这样的masas。迄今为止获得的结果表明该理论现在可以扩展到研究一般子代数。这与因子的结构（算子代数的构建模块）直接相关，这些拟议研究领域的总体目标是增加我们对该领域的理解。在先前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的对象，它们可以揭示同一底层对象的不同方面。熵是一个区分不同自同构的数值常数。提议者之前的联合工作表明，基于完全正映射的当前理论可以更好地根据完全收缩来重新表述。这使该理论处于更加灵活的境地，可以使用更通用和更强大的工具来解决该领域的问题，特别是与自同构群的交叉积相关的问题。 现代算子代数的研究有两个主要来源。矩阵是数字的概括，最初是为了求解方程而引入的，现在它的应用范围从计算机图形学到网络搜索引擎。在以数学方式表述量子力学时，冯·诺依曼发现他需要称为线性算子的矩阵的无限维版本，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叉积来表达，而拓扑熵在这里发挥了重要作用。该项目主要关注算子代数的理论基础，但这些领域拟议的工作可能会影响其中一些更具体的领域，因为有限因子是那些最接近模拟矩阵的算子代数。