We study the phase transition of KMS states for the C*-algebras of $ax+b$-semigroups of algebraic integers in which the multiplicative part is restricted to a congruence monoid, as in recent work of Bruce generalizing earlier work of Cuntz, Deninger, and Laca. Here we realize the extremal low-temperature KMS states as generalized Gibbs states by constructing concrete representations induced from extremal traces of certain group C*-algebras. We use these representations to compute the Murray--von Neumann type of extremal KMS states and we determine explicit partition functions for the type I factor states. The collection of partition functions that arise this way is an invariant under $\mathbb{R}$-equivariant isomorphism of C*-dynamical systems, which produces further invariants through the analysis of the topological structure of the KMS state space. As an application we characterize several features of the underlying number field and congruence monoid in terms of these invariants. In most cases our systems have infinitely many type I factor KMS states and at least one type II factor KMS state at the same inverse temperature and there are infinitely many partition functions. In order to deal with this multiplicity, we establish, in the context of general C*-dynamical systems, a precise way to associate partition functions to extremal KMS states that are of type I, and we then show that for our systems these partition functions depend only on connected components in the KMS simplex. The discussion of partition functions of general C*-dynamical systems may be of interest by itself and is likely to have applications in other contexts, so we include it in a self-contained initial section that is partly expository and is independent of the number-theoretic background and of the technical results about congruence monoids.

中文翻译：

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划分函数作为 C*-动态不变量和同余幺半群的作用

我们研究了代数整数的 $ax+b$-半群的 C*-代数的 KMS 状态的相变，其中乘法部分仅限于同余幺半群，正如布鲁斯最近的工作概括了 Cuntz、Deninger 的早期工作，和拉卡。在这里，我们通过构造从某些群 C*-代数的极值迹导出的具体表示，将极值低温 KMS 状态实现为广义 Gibbs 状态。我们使用这些表示来计算 Murray--von Neumann 类型的极值 KMS 状态，并确定类型 I 因子状态的显式分区函数。以这种方式产生的分区函数的集合是 C*-动力系统的 $\mathbb{R}$-等变同构下的不变量，它通过对 KMS 状态空间的拓扑结构的分析产生进一步的不变量。作为一个应用程序，我们根据这些不变量来表征底层数域和同余幺半群的几个特征。在大多数情况下，我们的系统在相同的逆温度下具有无限多个 I 型因子 KMS 状态和至少一个 II 型因子 KMS 状态，并且存在无限多个配分函数。为了处理这种多重性，我们在一般 C* 动态系统的上下文中建立了一种精确的方法来将分区函数与类型 I 的极值 KMS 状态相关联，然后我们证明对于我们的系统这些分区函数仅依赖于 KMS 单纯形中的连通分量。