Sums of Hermitian Operators and Connections to Connes' Embedding Problem; Hyperinvariant Subspaces

厄米算子之和以及与 Connes 嵌入问题的联系；

• 批准号: 0901220
• 负责人: Kenneth Dykema
• 金额: $ 24.52万
• 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: 2009
• 资助国家: 美国
• 起止时间: 2009-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0901220&HistoricalAwards=false
• 关键词: Sums; Hermitian; Operators; Connections; Connes

AbstractDykemaThe PI will investigate two fundamental problems in the theory of operators that are contained in II_1 factors. The first is Connes' embedding problem. Recent work of Collins and Dykema has shown that this problem is equivalent to a question about sums of operators in finite von Neumann algebras, and other recent work of Bercovici, Collins, Dykema, Li and Timotin has positively answered the first part of this question, showing that all Horn inequalities hold in all finite von Neumann algebras. The second problem is the hyperinvariant subspace problem. In particular, the PI will focus on the remaining open part of this problem for elements of II_1-factors, namely, the case of quasi-nilpotent operators in II_1-factors.Operators on infinite dimensional Hilbert space are used in mathematical models of quantum mechanics, and they are of significance in diverse areas of mathematics. We will work on two fundamental problems in operator theory: Connes' embedding problem and the hyperinvariant subspace problem. These concern different aspects of the structure of operators on infinite dimensional Hilbert spaces. We will focus on operators whose algebras possess traces. The first problem is about how well such operators can be approximated (in their mixed moments with respect to the trace) by operators on finite dimensional spaces. We will attack this problem by examining eigenvalues of sums of operators. The second problem is about the possibility of decomposing operators on infinite dimensional space by restricting them to invariant subspaces. In some recent progress, Haagerup and Schultz have proved the existence of such subspaces for a large class of operators, and we will focus on some specific operators for which this question is unresolved.

PI将研究包含在II_1因子中的算子理论中的两个基本问题。 第一个是Connes的嵌入问题。 最近的工作柯林斯和Dykema表明，这个问题是等价的问题，在有限的冯诺依曼代数和其他最近的工作，Bercovici，柯林斯，Dykema，李和Reintin肯定地回答了这个问题的第一部分，表明所有的霍恩不等式举行的所有有限冯诺依曼代数。 第二个问题是超不变子空间问题。 特别地，PI将集中于II_1-因子的元素的问题的剩余部分，即II_1-因子中的拟幂零算子的情况。无穷维Hilbert空间上的算子用于量子力学的数学模型中，它们在数学的各个领域都有重要意义。 我们将研究算子理论中的两个基本问题：Connes嵌入问题和超不变子空间问题。 这些涉及不同方面的结构运营商的无限维希尔伯特空间。 我们将集中讨论代数具有迹的算子。 第一个问题是关于如何以及这些运营商可以近似（在其混合时刻相对于跟踪）的运营商在有限维空间。 我们将通过考察算子和的本征值来解决这个问题。 第二个问题是关于无限维空间上的算子通过限制到不变子空间来分解的可能性。 在最近的一些进展中，Haagerup和Schultz已经证明了一大类算子存在这样的子空间，我们将专注于一些特定的算子，这个问题尚未解决。