Representations of Cuntz-Krieger relations, dynamics on Bratteli diagrams, and path-space measures
Representations of Cuntz-Krieger relations, dynamics on Bratteli diagrams, and path-space measures
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Cuntz-Krieger 关系的表示、Bratteli 图上的动力学以及路径空间测量
DOI：
10.1090/conm/650/13008
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
P. Jorgensen
中科院分区：
文献类型：
--
作者：
S. Bezuglyi;P. Jorgensen
我们研究一类新的表示的Cuntz-Krieger代数$\mathcal O_A$构造的半分支函数系统，自然有关的静态Bratteli图。定义并研究了同构半分支函数系的概念。我们证明了这类系统的任何同构都蕴涵着Cuntz-Krieger代数$\mathcal O_A$的相应表示的等价性。特别是，我们证明了等价的措施产生等价表示的$\mathcal O_A$。我们使用马尔可夫测度定义在平稳Bratteli图的路径空间上，构造了$\mathcal O_A$的同构表示。要做到这一点，我们将一个（强）有向图固定（简单）Bratteli图，并表明同构图生成同构半分支函数系统。我们还考虑一类幺一表示的Cuntz-Krieger代数，我们将它们分类为酉等价。几个例子说明的结果包括在文件中。
We study a new class of representations of the Cuntz-Krieger algebras $\mathcal O_A$ constructed by semibranching function systems, naturally related to stationary Bratteli diagrams. The notion of isomorphic semibranching function systems is defined and studied. We show that any isomorphism of such systems implies the equivalence of the corresponding representations of Cuntz-Krieger algebra $\mathcal O_A$. In particular, we show that equivalent measures generate equivalent representations of $\mathcal O_A$. We use Markov measures which are defined on the path space of stationary Bratteli diagrams to construct isomorphic representations of $\mathcal O_A$. To do this, we associate a (strongly) directed graph to a stationary (simple) Bratteli diagram, and show that isomorphic graphs generate isomorphic semibranching function systems. We also consider a class of monic representations of the Cuntz-Krieger algebras, and we classify them up to unitary equivalence. Several examples that illustrate the results are included in the paper.