Studies in Operator Algebras

提议者将继续研究长期存在的关于冯诺伊曼代数上同调的Kadison-Ringroseconstruction。他将使用和完善一些最近开发的技术需要显示某些多线性算子的完整有界性。调查的cohomologygroups自然导致最近推出的概念，赋范子代数，这一主题将继续，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，极大交换子代数（masas）和一般的子代数冯诺依曼代数将研究。在联合工作中，提议者引入了强奇异masas的概念，并证明了许多由双曲群（包括自由群的一类）产生的有限因子具有这样的masas。到目前为止得到的结果表明，该理论现在可以扩展到研究一般的子代数。这直接关系到因子的结构（算子代数的构建块），这些研究领域的总体目标是增加我们在这一领域的理解。在以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。熵是区分不同自同构的一个数值常数。提出者之前的联合工作表明，目前的理论，基于完全正映射，可以更好地重新表述为完全压缩。这使理论进入一个更加灵活的情况下，更普遍和强大的工具，可以承担的问题领域，特别是有关交叉产品的自同构群。 算子代数的现代研究有两个主要来源。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网络搜索引擎都有应用。在制定量子力学数学，冯诺依曼发现，他需要无限维版本的矩阵称为线性算子，最好的研究在算子代数。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，拓扑熵在这里起着重要的作用。该项目主要关注算子代数的理论基础，但这些领域的拟议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型矩阵的算子代数。

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.