Studies in Operator Algebras

作者将继续研究长期存在的关于von Neumann代数上同调的Kadison-Ringroond猜想。他将使用和改进一些最近开发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而然地引出了最近引入的赋范子代数的概念，由于它直接适用于上同调和有界投影问题，因此这一主题将被继续进行。在一个不同但相关的方向上，我们将研究von Neumann代数的极大交换子代数(MAS)和通用子代数。在联合工作中，作者引入了强奇异MAS的概念，并证明了许多由双曲群(包括自由群的一类)产生的有限因子都有强奇异MAS。到目前为止所得到的结果表明，该理论现在可以扩展到研究一般的子代数。这与因子的结构(算子代数的构件)直接相关，这些拟议研究领域的总体目标是增加我们在这一领域的理解。在前人工作的基础上，我们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，它们可以揭示同一基础对象的不同方面。熵是区分不同自同构的一个数值常数，前人的联合工作表明，当前基于完全正映射的理论可以更好地用完全压缩的形式来表示。这使该理论进入了一个更灵活的情况，在那里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积有关的问题。现代对算子代数的研究主要有两个来源。矩阵是数字的推广，最初是用来解方程的，现在从计算机图形学到网络搜索引擎都有应用。在建立量子力学的数学模型时，冯·诺伊曼发现他需要被称为线性算符的矩阵的无限维版本，而线性算符是在算符代数中最好的研究对象。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，而拓扑熵在其中起着重要的作用。该项目主要涉及算子代数的理论基础，但在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于建模矩阵的算子代数。

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.