论文标题
在$ c^*$ - 与转移操作员相关的代数
On $C^*$-algebras associated to transfer operators for countable-to-one maps
论文作者
论文摘要
我们的初始数据是转移操作员$ L $，用于连续的，可计数的地图$φ：δ\至x $在本地紧凑的Hausdorff Space $ x $的开放子集上定义的x $。然后，$ l $可以用“电势”来识别，即，除非$φ$是本地同构形态，否则不需要连续的地图$ \ varrho：δ\至x $。我们将交叉产品$ C_0（x）\ rtimes l $定义为带有明确发电机和关系的通用$ C^*$ - 代数，并给出了$ C_0（x）\ rtimes l $的明确忠实表示，根据加权构图运算符生成。我们解释了它与Exel-Royer的交叉产品的关系，Quiver $ C^*$ - Muhly和Tomforde的代数，$ C^*$ - 与Kajiwara和Watatani相关的与复杂或自相似动力学相关的代数，以及Groupoid $ C^*$ c^*$ - 与Deaconu-renault-Renault-Renault-Renault-renault-renault-groupsoid相关的代数。 我们描述了$ C_0（x）\ rtimes l $的核心子代理的光谱，并用它来表征$ C_0（x）\ rtimes l $的简单性，并证明了$ c_0（x）\ rtimes l $的唯一性定理。我们为$ C_0（x）\ rtimes l $提供有效的标准，为Kirchberg代数，我们讨论了$ C_0（x）\ rtimes l $的核心子代数上的KMS状态之间的关系。
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 $φ$.
