Mathematical Sciences: Structural Properties of Non-Selfadjoint Operator Algebras

[9500839]本文描述了对非自伴随算子代数的结构性质的研究。这是一个结合了旧的自伴随算子领域的代数见解和单算子理论技术的领域。这一建议的目的是在最近应用一种新的思想对该领域的重要和困难问题取得成功的基础上再接再厉。在过去的几年里，“无限多重代数”和“插值算子”这两个概念的相互作用导致了该领域许多长期存在的问题的解决。其结果是将一类非自伴随算子代数的认识提升到一个全新的水平。在个人工作中，一位研究者建议进一步研究和分类极大三角代数，而Pitts建议研究（并有望完成）分配交换子空间格代数的相似理论，并对这种类型的解析代数进行分类。算子可以看作是有限的或无限的复数矩阵。通过这种方式，运算符的集合通常可以被赋予一个代数结构，并称为一个运算符代数。这些算子代数的研究植根于数学物理。代数理论中的主要问题是对代数进行分类，以便识别等价或相似的代数。这个项目处理具有三角形元素（作为矩阵）的算子代数的分类问题。＊＊＊

9500839 Orr This proposal describes research into the structural properties of non-self-adjoint operator algebras. This is a field which combines the algebraic insights of the older field of self-adjoint operators, from which it grows, with the techniques of single operator theory. The aim in this proposal is to build on the success that recent applications of a new circle of ideas have had with important and difficult problems in the field. The last few years have seen the interplay of two ideas, "infinite multiplicity algebras" and "interpolating operators" lead to the solution of a number of longstanding problems in the field. The effect has been to open up one class of non-selfadjoint operator algebras to a whole new level of understanding. In individual work, one investigator proposes to further study and classify maximal triangular algebras, while Pitts proposes to study (and hopefully complete) the similarity theory for distributive commutative subspace lattice algebras and to classify the analytic algebras of this type. Operators can be viewed as finite of infinite matrices of complex numbers. In this way a collect of operators can often be given an algebraic structure and referred to as an operator algebra. The study of these algebras of operators is rooted in mathematical physics. The main problem in the theory of algebras is to classify these algebras so that equivalent or similar algebras can be recognized. This project deals with the classification problem for operator algebras that have elements (viewed as matrices) in triangular form. ***