Classification of non-simple purely infinite C*-algebras

某些C*-代数类可以用简单不变量来分类。这对于纯无限的、核的、可分的、单的C*-代数是众所周知的。允许非单C*-代数给分类问题增加了新的拓扑复杂性，这一点一般还没有被理解。在许多情况下，新发现不变量给出了完整的分类结果。特别地，这个新的不变量给出了图C*-代数和Hasse图为箭图的有限空间上的C*-代数的一个强分类定理。这个项目将进一步发展这种方法来对非简单C*-代数进行分类，特别是我们想要找出不变量在图C*-代数上可能具有哪些值，如何以及用什么数据来显式地计算不变量，以及如何将分类结果扩展到其他类别的例子。对于图C*-代数，我们想要实现一个计算机算法，利用不变量来判定两个图C*-代数是否同构。不变量还暗示了如何在不损失太多良好结构的情况下扩大图C*-代数的类。按照这些提示，可以找到可分类纯无限C*-代数的更一般的模型。

Certain classes of C*-algebras may be classified by simple invariants. This is well-known for purely infinite, nuclear, separable, simple C*-algebras. Allowing non-simple C*-algebras adds a new topological complexity to the classification problem, which is not yet understood in general. In many case, a newly found invariant gives complete classification results. In particular, this new invariant gives a strong classification theorem for graph C*-algebras and for C*-algebras over finite spaces whose Hasse diagram is a quiver. This project will further develop this approach to classification of non-simple C*-algebras.In particular, we want to find out which values the invariant may take on graph C*-algebras, how and with what data to compute the invariant explicitly, and how to extend the classification result to other classes of examples. For graph C*-algebras, we want to implement a computer algorithm that decides whether two graph C*-algebras are isomorphic using the invariant. The invariant also hints at how to enlarge the class of graph C*-algebras without losing too much of their nice structure. These hints shall be followed to find more general models for classifiable purely infinite C*-algebras.