数域上C* 代数-代数的相变与广义Furstenberg猜想
Phase transitions on C*-algebras arising from number fields and the generalized Furstenberg conjecture
摘要：
在最近的工作中，Cuntz，Deninger和Laca研究了数域中代数整数的仿射幺半群在由绝对范数决定的时间演化下的Toeplitz型C*-代数。系统的KMS平衡态由半直积$J \rtimes O^*$的C* 代数上的迹参数化，半直积是由单位$O^*$在代表每个理想类的积分理想$J$上的乘法作用产生的.在每个固定的逆温度$β> 2$处，极端平衡状态对应于$C^*(J\rtimes O^*)$的极端迹线。在这里，我们进行研究这些痕迹使用转置行动$O^*$上的$\hat J$的理想和最近表征的痕迹上的变换群C* 代数由于Neshveyev。我们表明，极值轨迹的$C^*(J\rtimes O^*)$参数化的对组成的遍历不变的措施的行动$O^*$ $\hat{J}$连同一个字符的各向同性子组相关联的支持这一措施。对每一个理想，对偶群$\hat {J}$是一个d-环面，$O^*$通过线性环面自同构作用于其上。因此，分类所有极值迹的问题是Furstenberg著名的$\times 2$ $\times 3$猜想的推广版本。我们分类的结果，为各种数字领域的理想类组，程度，和单位秩，我们指出了一路上的平凡，棘手的，和可分类的情况下。在拓扑水平上，它是可能的，以表征数域的无限$O^*$不变集是密集的$\hat{J} $，由于定理的Berend;作为一个应用程序，我们给出了描述的原始理想空间的$C^*(J\rtimes O^*)$为这些数域。
英文摘要：
In recent work, Cuntz, Deninger and Laca have studied the Toeplitz type C*-algebra associated to the affine monoid of algebraic integers in a number field, under a time evolution determined by the absolute norm. The KMS equilibrium states of their system are parametrized by traces on the C*-algebras of the semidirect products $J \rtimes O^*$ resulting from the multiplicative action of the units $O^*$ on integral ideals $J$ representing each ideal class. At each fixed inverse temperature $β> 2$, the extremal equilibrium states correspond to extremal traces of $C^*(J\rtimes O^*)$. Here we undertake the study of these traces using the transposed action of $O^*$ on the duals $\hat J$ of the ideals and the recent characterization of traces on transformation group C*-algebras due to Neshveyev. We show that the extremal traces of $C^*(J\rtimes O^*)$ are parametrized by pairs consisting of an ergodic invariant measure for the action of $O^*$ on $\hat{J}$ together with a character of the isotropy subgroup associated to the support of this measure. For every ideal the dual group $\hat {J}$ is a d-torus on which $O^*$ acts by linear toral automorphisms. Hence, the problem of classifying all extremal traces is a generalized version of Furstenberg's celebrated $\times 2$ $\times 3$ conjecture. We classify the results for various number fields in terms of ideal class group, degree, and unit rank, and we point along the way the trivial, the intractable, and the conjecturally classifiable cases. At the topological level, it is possible to characterize the number fields for which infinite $O^*$-invariant sets are dense in $\hat{J} $, thanks to a theorem of Berend; as an application we give a description of the primitive ideal space of $C^*(J\rtimes O^*)$ for those number fields.
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