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; 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.

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,无限维，复杂的希尔伯特空间。在B（h）中，已经发现了一类双面理想的某些代数K组之间的密切联系。在表征那些具有零同源性的理想（即那些是理想本身的换向者理想，因此没有痕迹的换向者理想）以及概括N.J. Kalton对Hilbert-Schmidt班级的换向者理想的表征，以及Trace comporter of Trace Class（b（H）到仲裁理想的换向者理想的情况，这是最新的重要进步（即那些是理想本身，因此没有痕迹的换向理想）。自然的概括是对那些换向者理想（b（h））的理想的特征，其中包含给定紧凑型操作员T的t。 对于对称规范的理想，具有密集生成的次视野的对称规范理想的答案已经解决。正在开发的技术涉及利用最近发现的障碍，以防止某些理想的痕量扩展超出某些较小的理想，并且与相关的换向器类别和交换器理想有关。 对运营商代数和操作员理想的研究（即，特殊类别的操作员的每个操作员可以将其视为无限矩阵（平方，无限数字阵列））涉及对其操作员之间的操作研究，尤其是加法和乘法。这些操作的操作员代数在整个数学，科学，商业，计算机科学及其在其他F IELD上的应用中都有必不可少的滚动。众所周知的乘法定律是，对于所有数字A和B的AB = BA，即，无论哪种订单乘以数字，一个人总是会获得相同的结果。具有该特性的元素的代数称为交换代数。但是，大多数代数并不是交换性的，并且研​​究其非零换向因子AB-BA的结构在理解代数及其与更容易理解的交换代数的关系方面成为核心，而AB-BA始终为零。著名的海森伯格物理学中不确定的原则提供了换向者相关性的早期例子。该建议是扩大我们对许多重要操作员代数的换向器结构的了解，并利用新信息来开发技术，以解决通勤理论以及操作员代数和理想中的一些新旧问题。