关于与可数对一映射的转移算子相关的 $C^*$-代数
On $C^*$-algebras associated to transfer operators for countable-to-one maps
AI总结：
本文为可数对一映射的转移算子构造泛交叉积 $C_0(X)\rtimes L$，给出加权复合算子表示，并研究其与已知 $C^*$-代数的关系、单性、唯一性、Kirchberg 判据及 KMS 态。
摘要：
我们的初始数据是针对连续的可数对一映射 $φ:Δ\to X$ 的转移算子 $L$，该映射定义在局部紧 Hausdorff 空间 $X$ 的一个开子集上。于是 $L$ 可以等同于一个“势”，即映射 $\varrho:Δ\to X$；除非 $φ$ 是局部同胚，否则它不必连续。我们将交叉积 $C_0(X)\rtimes L$ 定义为具有显式生成元和关系的泛 $C^*$-代数，并给出 $C_0(X)\rtimes L$ 的显式忠实表示，在该表示下它由加权复合算子生成。我们阐明了它与 Exel-Royer 交叉积、Muhly 和 Tomforde 的箭图 $C^*$-代数、Kajiwara 和 Watatani 与复动力学或自相似动力学相关的 $C^*$-代数，以及与 Deaconu-Renault 群胚相关的群胚 $C^*$-代数之间的关系。我们描述了 $C_0(X)\rtimes L$ 的核心子代数的谱，并用其刻画 $C_0(X)\rtimes L$ 的单性，证明了 $C_0(X)\rtimes L$ 的唯一性定理。我们给出了 $C_0(X)\rtimes L$ 成为 Kirchberg 代数的有效判据，并讨论了 $C_0(X)\rtimes L$ 核心子代数上的 KMS 态与 $φ$ 的共形测度之间的关系。
英文摘要：
Our initial data is a transfer operator $L$ for a continuous, countable-to-one map $φ:Δ\to X$ defined on an open subset of a locally compact Hausdorff space $X$. Then $L$ may be identified with a `potential', i.e. a map $\varrho:Δ\to X$ that need not be continuous unless $φ$ is a local homeomorphism. We define the crossed product $C_0(X)\rtimes L$ as a universal $C^*$-algebra with explicit generators and relations, and give an explicit faithful representation of $C_0(X)\rtimes L$ under which it is generated by weighted composition operators. We explain its relationship with Exel-Royer's crossed products, quiver $C^*$-algebras of Muhly and Tomforde, $C^*$-algebras associated to complex or self-similar dynamics by Kajiwara and Watatani, and groupoid $C^*$-algebras associated to Deaconu-Renault groupoids. We describe spectra of core subalgebras of $C_0(X)\rtimes L$ and use it to characterise simplicity of $C_0(X)\rtimes L$ and prove the uniqueness theorem for $C_0(X)\rtimes L$. We give efficient criteria for $C_0(X)\rtimes L$ to be a Kirchberg algebra, and we discuss relationship between KMS states on the core subalgebra of $C_0(X)\rtimes L$ and conformal measures for $φ$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
