We investigate almost minimal actions of abelian groups and their crossed products. As an application, given multiplicatively independent integers p and q, we show that Furstenberg’s $\times p,\times q$ conjecture holds if and only if the canonical trace is the only faithful extreme tracial state on the $C^*$-algebra of the group $\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2$. We also compute the primitive ideal space and K-theory of $C^*(\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2)$.

中文翻译：

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弗斯滕伯格猜想的痕迹表征

我们研究了阿贝尔群及其交叉乘积的几乎最小行为。作为一个应用，给定乘法独立的整数p和q，我们证明 Furstenberg 的$\times p,\times q$猜想成立当且仅当规范迹是$C^*$代数上唯一忠实的极端迹状态群$\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2$。我们还计算了$C^*(\mathbb {Z}[\frac {1}{pq}]\rtimes \mathbb {Z}^2)$的原始理想空间和 K 理论。