Geometric Classification of Graph C*-algebras over Finite Graphs
Geometric Classification of Graph C*-algebras over Finite Graphs
复制标题
有限图上图 C* 代数的几何分类
DOI：
10.4153/cjm-2017-016-7
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
Adam P. W. Sørensen
中科院分区：
文献类型：
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作者：
S. Eilers;Gunnar Restorff;Efren Ruiz;Adam P. W. Sørensen
摘要 我们解决图 ${{C}^{*}}$ 有限图（有限多个边和顶点）的代数的分类问题，其中包含 Cuntz-Krieger 代数类作为突出的特例。与早期的工作相比，我们并不假设图满足标准条件 $\left( K \right)$ ，因此图 ${{C}^{*}}$ 代数可能带有无数个理想。我们发现，在这种普遍性中，图 ${{C}^{*}}$ -代数的稳定同构与 Cuntz 移动等价的几何概念并不相符。然而，在图上添加一个适度的条件，这两个概念被证明是相互等价的，并且等价于具有同构 $K$ 理论的 C* 代数。这反过来证明，在这种情况下，图 ${{C}^{*}}$ 代数实际上可以通过 $K$ 理论进行分类，特别是当所讨论的 ${{C}^{*}}$ 代数是实秩零或类型 I/后阈值时，提供完整的分类。获得这些结果的关键因素是使用图的邻接矩阵来表征 Cuntz 移动等价性。我们的结果应用于讨论 Hong 和 Szymański 定义的量子透镜空间的分类问题，并完成与所有具有四个或更少顶点的简单图相关的图 ${{C}^{*}}$ 代数的分类。
Abstract We address the classification problem for graph ${{C}^{*}}$ -algebras of finite graphs (finitely many edges and vertices), containing the class of Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work, we do not assume that the graphs satisfy the standard condition $\left( K \right)$ , so that the graph ${{C}^{*}}$ -algebras may come with uncountably many ideals. We find that in this generality, stable isomorphism of graph ${{C}^{*}}$ -algebras does not coincide with the geometric notion of Cuntz move equivalence. However, adding a modest condition on the graphs, the two notions are proved to be mutually equivalent and equivalent to the C*-algebras having isomorphic $K$ -theories. This proves in turn that under this condition, the graph ${{C}^{*}}$ -algebras are in fact classifiable by $K$ -theory, providing, in particular, complete classification when the ${{C}^{*}}$ - algebras in question are either of real rank zero or type I/postliminal. The key ingredient in obtaining these results is a characterization of Cuntz move equivalence using the adjacency matrices of the graphs. Our results are applied to discuss the classification problem for the quantumlens spaces defined by Hong and Szymański, and to complete the classification of graph ${{C}^{*}}$ -algebras associated with all simple graphs with four vertices or less.