Mathematical Sciences: Semigroup of Endomorphisms of Operator Algebras

9500291这里提出的研究是关于算子代数的自同构半群的结构和分类，以及它们与现代分析、量子物理和概率论的几个领域的关系的问题。这个项目的一个主要组成部分是在某些半群和Hilbert空间的连续张量积之间建立联系。更确切地说，E_0-半群直到上循环共轭的分类等价于对这些连续张量积系统的分类问题。反过来，后一种结构产生了各种类型的可计算不变量：数值(指数不变量)、拓扑(非对易拓扑空间，谱C*-代数)。此外，这种连续张量积在概率论中是自然出现的；例如，它们可以与具有平稳独立增量的随机分布相关联，也可以与其他鞅类型的过程相关联。第二个项目是继续作者最近关于算子代数在计算无限维自伴算子谱问题中的作用的工作。这个项目的一般数学领域以希尔伯特空间上的算子代数理论为基础。运算符可以被认为是复数的有限或无限矩阵。特殊类型的运算符通常放在一个代数中，自然称为运算符代数。这些看似抽象的对象有着令人惊讶的各种应用。例如，它们在纽结理论中扮演着关键角色，而纽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. ***