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

抽象的韦斯 算子理想的交换子结构（例如，单交换子类AB-BA，其中A，B分别来自两个双边算子理想，沿着交换子理想（它们的线性跨度））和相关的迹扩张问题将被研究和利用，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、无限维的复Hilbert空间上的有界线性算子类。对于B（H）中的一类广泛的双边理想，发现了某些代数K-群之间的密切联系。最近的重要进展已经在表征那些具有零同调的理想（即，这些交换子理想是理想本身，因此没有迹）和推广N。J. Kalton对希尔伯特-施密特类的换位子理想以及迹类的换位子理想（具有B（H））的刻画到任意理想。一个自然的推广是对于给定的紧算子T，其交换子理想（与B（H））包含T的理想的特征。在这方面出现的一个问题是：哪些赋范理想承认一个连续的或积极的痕迹？ 这个问题的答案是对称赋范理想与一个稠密的生成子理想已经解决。正在开发的技术涉及利用最近发现的障碍，防止某些理想具有超出某些较小理想的迹扩展，并且与相关的换位子类和换位子理想有关。 算子代数和算子理想的研究（即，特殊类别的运算符，其中每个运算符可以被视为一个无限矩阵（一个正方形，无限数组的数字））涉及调查的操作之间的操作，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、计算机科学以及它们在其他领域的许多应用中起着不可或缺的作用。众所周知的乘法交换律是AB = BA对所有数A和B，即，无论哪一阶相乘，结果总是一样的。元素具有这种性质的代数称为交换代数。但是大多数代数不是交换的，研究它们的非零交换子AB-BA的结构成为理解代数的核心 以及它们与更容易理解的交换代数的关系，在交换代数中AB-BA总是零。物理学中著名的海森伯不确定性原理提供了一个早期的例子，证明了量子力学的相关性。这项建议是扩大我们的知识的换向器结构的一些重要的算子代数和使用新的信息对发展技术解决一些老的和新的开放问题的换向器理论和算子代数和理想。

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.