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

我们将研究和利用算子理想的交换子结构(例如，A、B分别来自两个双边算子理想的单交换子AB-BA以及交换子理想(它们的线性跨度))以及相关的迹扩张问题，以确定B(H)中双边理想的Hochschild和循环同调。B(H)是作用在可分的无限维复Hilbert空间上的有界线性算子类。对于B(H)中一大类双边理想，发现了某些代数K-群之间的密切联系。最近在刻划零同调的理想(即那些作为理想本身因而没有迹的交换子理想)以及将N·J.Kalton对Hilbert-Schmidt类的交换子理想的刻画以及迹类(具有B(H))的交换子理想的刻画推广到任意理想方面取得了重要进展。自然推广是交换子理想(具有B(H))对于给定紧算子T包含T的那些理想的刻画。由此产生的一个问题是：哪些赋范理想允许连续或正迹？对于具有稠密有限生成子理想的对称赋范理想，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障碍来阻止某些理想具有超出某些较小理想的迹扩展，并且与相关的交换子类和交换子理想相关。研究算子代数和算子理想(即，特殊类型的算子，其每个算子可以被视为一个无限矩阵(一个正方形、无限数组))涉及研究它们的算子之间的运算，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，对于所有的数A和B，AB=BA，也就是说，无论哪一阶乘数，都得到相同的结果。元素具有这一性质的代数称为交换代数。但大多数代数都不是交换的，研究它们的非零交换子AB-BA的结构成为理解这些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总是零的。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。这一建议是为了扩大我们对一些重要的算子代数的交换子结构的认识，并利用新的信息来发展技术来解决交换子理论、算子代数和理想中的一些新的和旧的公开问题。

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.