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

[503062] Weiss本课题的主要目标是确定有界Hilbert空间算子的全代数相对于任意二边理想的Hochschild和循环上同群。这些计算最近与对易子理想的表征联系在一起。迹扩展和对易子和对易子空间的结构将被研究和利用，以产生关于算子理想的同调、行列式和k理论的信息。这个项目位于两个数学领域的交汇处。一方面，该项目涉及运算符的交换子的精确显式计算。算符理论是作为数学物理方程的抽象而发展起来的。换向子对应于有序观测之间的差。在另一个方向上，运算符的集合通常可以被认为是称为运算符代数的代数结构。这些代数的某些不变量在包括数学、物理和几何在内的几个学科中起着重要的作用。这个项目的目的是通过执行精细的换向子计算来实现这些不变量。＊＊＊

9503062 Weiss A principal goal of this project is to determine the Hochschild and cyclic cohomology groups for the full algebra of bounded Hilbert space operators relative to an arbitrary 2-sided ideal. These computations have been recently linked to characterizing commutator ideals. Trace extensions and the structure of commutators and commutator spaces will be investigated and exploited to yield information on the homology, determinants and K-theory of operator ideals. This project lies at the interface of two mathematical areas. On one side, the project concerns delicate explicit computations of commutators of operators. Operator theory evolved as an abstraction of the equations of mathematical physics. A commutator corresponds to the difference between ordered observations. In another direction, a collection of operators can often be considered as an algebraic structure called an operator algebra. Certain invariants of such algebras play an important role in several disciplines including mathematical physics and geometry. The purpose of this project is to realize these invariants by performing delicate commutator computations. ***