Mathematical Sciences: Semigroup of Endomorphisms of Operator Algebras

本文提出的研究涉及算子代数自同构半群的结构和分类问题，以及它们与现代分析、量子物理和概率论等几个领域的关系。该项目的一个主要组成部分是建立某些半群与希尔伯特空间的连续张量积之间的联系。更准确地说，e_0 -半群直至循环共轭的分类问题等价于这些连续张量积系统的分类问题。反过来，后一种结构产生各种类型的可计算不变量：数值不变量（索引不变量），拓扑不变量（非交换拓扑空间，谱C*-代数）。而且，这种连续张量积在概率论中自然存在；例如，它们可以与具有平稳独立增量的随机分布相关联，也可以与其他鞅型过程相关联。第二个项目是继续提出者最近的工作，在计算无穷维自伴随算子的谱问题上算子代数的作用。这个项目的一般数学领域在希尔伯特空间上的算子代数理论中有它的基础。运算符可以被认为是有限或无限的复数矩阵。特殊类型的运算符通常放在一个代数中，自然地称为运算符代数。这些看似抽象的对象有着各种各样的应用。例如，它们在结理论中起着关键作用，而结理论目前正被用于研究DNA的结构。＊＊＊

9500291 Arveson The research proposed here addresses problems related to the structure and classification of semigroups of automorphisms of operator algebras and their relationship to several areas of modern analysis, quantum physics, and probability theory. A principal component of the project is to establish a connection between certain semigroups and continuous tensor products of Hilbert spaces. more precisely, the classification of E_0-semigroups up to cocycle conjugacy was shown to be equivalent to the problem of classifying these continuous tensor product systems. In turn, the latter structures give rise to computable invariants of various types: numerical (an index invariant), topological (a non-commutative topological space, the spectral C*-algebra). Moreover, such continuous tensor products occur naturally in probability theory; for example, they can be associated with random distributions having stationary independent increments, and with other martingale-type processes as well. A secondary project is to continue the proposer's recent work on the role of operator algebras in the problem of computing the spectra of infinite dimensional self-adjoint operators. The general area of mathematics of this project has its basis in the theory of algebras of operators on Hilbert space. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These seemingly abstract objects have a surprising variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA. ***