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

摘要：本文研究了算子理想的换位子结构（例如，单换位子AB-BA类，分别具有A和B的两个双边算子理想，以及换位子理想（它们的线性张成））和相关的迹扩展问题，以确定B(H)中双边理想的Hochschild和循环同调，B(H)是作用于可分离的无限维复Hilbert空间的有界线性算子类。在B(H)中，对于一类广义的双边理想，我们发现了若干代数k群之间的密切联系。最近在描述零同调的理想（即，那些交换子理想本身就是理想，因此没有迹）和推广N. J. Kalton对Hilbert-Schmidt类的交换子理想的描述，以及对迹类的交换子理想（带B(H)）的描述到任意理想方面取得了重要进展。自然推广是对那些理想的表征，这些理想的对易子理想（带B(H)）对于给定的紧算子T包含T。在这种情况下产生的问题是：哪些赋范理想允许连续或正迹？对于具有密集有限生成子粒子的对称赋范理想，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阻止某些理想具有超越某些较小理想的跟踪扩展的障碍，并且与相关的换向子类和换向子理想相关。算子代数和算子理想（即一类特殊的算子，其中的每个算子都可以看作是一个无限矩阵（一个正方形，无限数组））的研究涉及到算子之间的运算，特别是加法和乘法的研究。具有这些运算的算子代数在数学、科学、商业、计算机科学及其在许多其他领域的应用中发挥着不可或缺的作用。众所周知的乘法交换律是AB = BA对所有的数字A和B，也就是说，一个人总是得到相同的结果，无论你乘哪个顺序的数字。元素具有这种性质的代数称为交换代数。但大多数代数是不可交换的，研究它们的非零对易子AB-BA的结构，对于理解代数及其与更容易理解的交换代数的关系至关重要，在交换代数中AB-BA总是零。物理学中著名的海森堡测不准原理为换向子的相关性提供了一个早期的例子。本文的目的是扩大我们对一些重要算子代数的换向子结构的认识，并利用新的信息发展技术来解决换向子理论和算子代数及理想中的一些新的和旧的开放性问题。

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.