Mathematical Sciences: Infinite Simple C*-Algebras and TheirInvariants

在接下来的三年里，首席研究员张爽建议继续研究以下三个问题。(1)是否存在一个简单的C*-代数，同时包含有限和无限的非零投影？(2)确定了由循环群的自由积和某些投影相关的约简群C*-代数生成的简单C*-代数，以及由加权双边移位和等距生成的简单C*-代数的k -理论。(3)将这些无核的、纯无限的、简单的C*-代数划分到同构。问题(1)是简单C*-代数理论和可分离简单C*-代数分类程序中的一个突出问题。作者打算用相关乘数代数中的一些技术来解决这个问题。(2)和(3)中的C*-代数包括非核的、纯无限的、简单的C*-代数，它们可以作为某些非核简单C*-代数的标准模型。对这些C*-代数结构的研究将补充扩充分类程序到无核简单C*-代数的范畴。受几位主要研究人员对上述项目的兴趣的鼓舞，提议者期望这项调查将激发有关主题的富有成效的研究活动。三维空间和任何有限维空间中的所有线性变换的集合是由平移和某些旋转（通过加法、标量乘法和复合）生成的。这个集合是简单C*-代数的基本例子，可以用所有方阵的代数来标识。这些初等简单C*-代数是由空间维数完全分类的。量子物理和现代技术中的许多问题（例如，计算机编程）都归结为对这类简单C*代数的研究，更一般地说，是对无限维空间中由某些线性变换产生的各种简单C*代数的研究。所提出的项目涉及由无限大小的方阵生成的简单C*-代数的结构。在增加对这些简单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。该计划特别关注研究区分这些代数的不变量。经典维数不再是区分不变量，因此寻找和研究新的区分不变量成为两个关键问题。拟议的项目旨在解决这一领域的这些突出问题。

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.