Let $(G,\unicode[STIX]{x1D6EC})$ be a self-similar $k$-graph with a possibly infinite vertex set $\unicode[STIX]{x1D6EC}^{0}$. We associate a universal C*-algebra ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$ to $(G,\unicode[STIX]{x1D6EC})$. The main purpose of this paper is to investigate the ideal structures of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. We prove that there exists a one-to-one correspondence between the set of all $G$-hereditary and $G$-saturated subsets of $\unicode[STIX]{x1D6EC}^{0}$ and the set of all gauge-invariant and diagonal-invariant ideals of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. Under some conditions, we characterize all primitive ideals of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. Moreover, we describe the Jacobson topology of some concrete examples, which includes the C*-algebra of the product of odometers. On the way to our main results, we study self-similar $P$-graph C*-algebras in depth.

中文翻译：

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自相似图C*-代数的理想结构

让$(G,\unicode[STIX]{x1D6EC})$自相似$k$- 具有可能无限顶点集的图$\unicode[STIX]{x1D6EC}^{0}$. 我们关联一个通用的 C*-代数${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$到$(G,\unicode[STIX]{x1D6EC})$. 本文的主要目的是研究理想的结构${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. 我们证明了所有的集合之间存在一一对应的关系$G$- 遗传和$G$-饱和的子集$\unicode[STIX]{x1D6EC}^{0}$和所有规范不变和对角不变理想的集合${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. 在某些条件下，我们刻画了所有的原始理想${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$. 此外，我们描述了一些具体例子的 Jacobson 拓扑，其中包括里程表乘积的 C*-代数。在获得主要结果的路上，我们研究了自相似$P$- 深度图 C*-代数。