Given a unital inductive limit of C*-algebras for which each C*-algebra of the inductive sequence comes equipped with a Rieffel compact quantum metric, we produce sufficient conditions to build a compact quantum metric on the inductive limit from the quantum metrics on the inductive sequence by utilizing the completeness of the dual Gromov–Hausdorff propinquity of Latrémolière on compact quantum metric spaces. This allows us to place new quantum metrics on all unital approximately finite-dimensional (AF) algebras that extend our previous work with Latrémolière on unital AF algebras with faithful tracial state. As a consequence, we produce a continuous image of the entire Fell topology on the ideal space of any unital AF algebra in the dual Gromov–Hausdorff propinquity topology.

中文翻译：

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C*-代数和紧量子度量空间的归纳极限

给定 C*-代数的单位归纳极限，其中归纳序列的每个 C*-代数都配备了 Rieffel 紧致量子度量，我们从通过利用 Latrémolière 的对偶 Gromov-Hausdorff 相似性在紧凑量子度量空间上的完整性进行归纳序列。这使我们能够在所有单位近似有限维 (AF) 代数上放置新的量子度量，这扩展了我们之前与 Latrémolière 在具有忠实 trac 状态的单位 AF 代数上的工作。因此，我们在对偶 Gromov-Hausdorff 邻近拓扑中的任何单位 AF 代数的理想空间上生成了整个 Fell 拓扑的连续图像。