Mathematical Sciences: Infinite Simple C*-Algebras and TheirInvariants

数学科学：无限简单 C* 代数及其不变量

• 批准号: 9626592
• 负责人: Shuang Zhang
• 金额: --
• 依托单位: University of Cincinnati Main Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626592&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Infinite; Simple; Algebras

DMS-9626592 PI: Shuang Zhang Univ of Cincinnati During the next three years the Principal Investigator, Shuang Zhang, proposes to continue his research on the following three problems. (1) Is there a simple C*-algebra that contains both finite and infinite nonzero projections? (2) Determine the K-theory of the simple C*-algebras generated by the reduced group C*-algebras associated with free products of cyclic groups and certain projections, and simple C*-algebras generated by a weighted bilateral shift and an isometry. (3) Classifying these non-nuclear, purely infinite, simple C*-algebras up to isomorphism. The problem (1) is an outstanding problem in the theory of simple C*-algebras and the program of classifying separable simple C*-algebras. The proposer intends to attack this problem by some techniques in the associated multiplier algebras. The C*-algebras in (2) and (3) include non-nuclear, purely infinite, simple C*-algebras which might serve as standard models in certain classes of non-nuclear simple C*-algebras. Investigation on the structure of these C*-algebras will supplement the expansion of classification program to the category of non-nuclear simple C*-algebras. Encouraged by the interests of several leading researchers on the projects above, the proposer expects that this investigation will stimulate fruitful research activities on the subjects. The collection of all linear transformations in the three dimensional space, and in any finite dimensional space, is generated by translations and certain rotations (via addition, scalar multiplication, and composition). This collection is the basic example of simple C*-algebras and can be identified with the algebra of all square matrices. These elementary simple C*-algebras are fully classified by the dimension of the space. Many problems in the quantum physics and in modern technologies (for example, computer programming) are reduced to the investigation of simple C*-algebras of this sort, and more generally, a variety of simple C*-algebras generated by certain linear transformations in the infinite dimensional space. The proposed project is concerned with the structure of simple C*-algebras generated by some square matrices of infinite size. In the process of increasing knowledge about these simple C*-algebras, one important step is to understand their structure and then to classify them. The proposed project is especially concerned with studying invariants that distinguish these algebras. The classical dimension is no longer the distinguishing invariant, and thus, finding and studying new distinguishing invariants become two key problems. The proposed project aims at attacking these outstanding problems in this area.

DMS-9626592 PI：辛辛那提的Shuang Zhang Univ在接下来的三年中，首席研究员Shuang Zhang提议继续对以下三个问题进行研究。 （1）是否有一个简单的c* - 代数包含有限和无限的非零投影？ （2）确定由降低的组C* - 代数与循环基团和某些投影相关的简单C* - 代数的K理论，以及由加权双侧移位和等轴测产生的简单C* - 代数。 （3）将这些非核，纯粹的无限，简单的C* - 代数分类为同构。 问题（1）是简单c* - 代数理论的一个杰出问题，也是对可分离的简单c*-ergebras进行分类的计划。提议者打算通过相关的乘数代数中的某些技术来攻击此问题。 （2）和（3）中的C* - 代数包括非核的，纯粹的无限，简单的C* - 代数，这些代数可能是某些类别的非核简单C*-ergebras中的标准模型。 对这些C*代数的结构的调查将补充分类程序的扩展到非核简单C* - 代数的类别。 在几位主要研究人员对上述项目的利益的鼓励下，提议者预计，这项调查将刺激对受试者的富有成果的研究活动。 在三维空间以及任何有限维空间中的所有线性转换的收集都是由翻译和某些旋转（通过加法，标量乘法和组成）生成的。该集合是简单C* - 代数的基本示例，可以用所有平方矩阵的代数识别。这些基本简单的c* - 代数由空间的维度完全分类。 量子物理学和现代技术（例如，计算机编程）中的许多问题都减少为对这种简单的C* - 代数的研究，更普遍地，由无限尺寸空间中某些线性转换产生的各种简单的C* - 代数。 拟议的项目涉及一些无限大小的平方矩阵产生的简单C* - 代数的结构。在增加有关这些简单的C* - 代数的知识的过程中，一个重要的步骤是了解它们的结构，然后对它们进行分类。拟议的项目特别关注研究区分这些代数的不变性。 经典维度不再是区别不变的，因此，发现和研究新的区分不变式成为两个关键问题。拟议的项目旨在攻击该领域的这些出色问题。