Mathematical Sciences: Trace Extensions, Commutator Spaces and Single Commutators with Applications to the Homology, Determinants and K-Theory of Operator Ideals

数学科学：迹扩展、换向器空间和单换向器及其在算子理想的同调性、行列式和 K 理论中的应用

• 批准号: 9706911
• 负责人: Gary Weiss
• 金额: --
• 依托单位: University of Cincinnati Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706911&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Trace; Extensions; Commutator

Abstract Weiss The commutator structure of operator ideals (e.g., the class of single commutators, AB-BA, with A, B from two two-sided operator ideals, respectively, along with commutator ideals (their linear span)) and related trace extension problems will be investigated and exploited to determine the Hochschild and cyclic homology for two-sided ideals in B(H), the class of bounded linear operators acting on a separable, infinite-dimensional, complex Hilbert space. A close connection between certain algebraic K-groups has been found for a wide class of two-sided ideals in B(H). Recent important advances have been made in characterizing those ideals which have zero homology (i.e., those commutator ideals which are the ideals themselves and hence have no trace) and in generalizing N. J. Kalton's characterization of the commutator ideal of the Hilbert-Schmidt class, and also the commutator ideal of the trace class (with B(H)), to arbitrary ideals. A natural generalization is a characterization of those ideals whose commutator ideal (with B(H)) contains T for a given compact operator T. A question arising in this context is: Which normed ideals admit a continuous or positive trace? The answer to this question for symmetrically normed ideals with a dense finitely generated subideal is already settled. Techniques under development involve exploiting recently found obstructions preventing certain ideals from having a trace extension beyond certain smaller ideals, and are related to the associated commutator classes and commutator ideals. The study of operator algebras and operator ideals (i.e., special classes of operators each operator of which may be viewed as an infinite matrix (a square, infinite array of numbers)) involves the investigation of operations between their operators, especially addition and multiplication. Operator algebras with these operations play an indispensable roll throughout mathematics, science, business, computer science and many of their applications to other f ields. The well-known commutative law of multiplication is that AB = BA for all numbers A and B, i.e., one always obtains the same result no matter which order one multiplies numbers. Algebras whose elements have this property are called commutative algebras. But most algebras are not commutative and studying the structure of their non-zero commutators, AB-BA, becomes central in understanding the algebras and their relation to the more easily understood commutative algebras, inside which AB-BA is always zero. The famous Heisenberg Uncertainly Principle in physics provides an early example of the relevance of commutators. This proposal is to expand our knowledge of the commutator structure of a number of important operator algebras and to use the new information towards developing techniques to settle some old and new open problems in commutator theory and in operator algebras and ideals.

摘要 Weiss 将研究和利用算子理想的换向器结构（例如，单换向器类 AB-BA，其中 A、B 分别来自两个两侧算子理想，以及换向器理想（它们的线性跨度））和相关的迹扩展问题，以确定 B(H) 中两侧理想的 Hochschild 和循环同源性，该类有界线性算子作用于可分离的、 无限维、复希尔伯特空间。对于 B(H) 中的一类广泛的双边理想，某些代数 K 群之间存在密切联系。最近在表征那些具有零同源性的理想（即那些本身就是理想因此没有迹的换向器理想）以及将 N. J. Kalton 对希尔伯特-施密特类换向器理想以及迹类换向器理想（具有 B(H)）的表征推广到任意理想方面取得了重要进展。自然概括是那些理想的表征，对于给定的紧算子 T，其换向器理想（具有 B(H)）包含 T。在这种情况下出现的问题是：哪些规范理想允许连续或正迹？ 对于具有密集有限生成子理想的对称范数理想，这个问题的答案已经确定。正在开发的技术涉及利用最近发现的阻碍某些理想具有超出某些较小理想的迹线延伸的障碍物，并且与相关的换向器类别和换向器理想相关。 算子代数和算子理想（即特殊类型的算子，其中每个算子都可以被视为无限矩阵（一个正方形的、无限的数字数组））的研究涉及对算子之间运算的研究，特别是加法和乘法。具有这些运算的算子代数在数学、科学、商业、计算机科学及其在其他领域的许多应用中发挥着不可或缺的作用。众所周知的乘法交换律是，对于所有数字 A 和 B，AB = BA，即无论以哪一种顺序相乘，总是得到相同的结果。元素具有这种性质的代数称为交换代数。但大多数代数不是可交换的，研究其非零交换子 AB-BA 的结构成为理解代数及其与更容易理解的交换代数的关系的核心，其中 AB-BA 始终为零。物理学中著名的海森堡不确定原理提供了换向器相关性的早期例子。该建议旨在扩展我们对一些重要算子代数的换向器结构的了解，并利用新信息来开发技术来解决换向器理论以及算子代数和理想中的一些新旧开放问题。