We explore the recently introduced local-triviality dimensions by studying gauge actions on graph $C^*$-algebras, as well as the restrictions of the gauge action to finite cyclic subgroups. For $C^*$-algebras of finite acyclic graphs and finite cycles, we characterize the finiteness of these dimensions, and we further study the gauge actions on many examples of graph $C^*$-algebras. These include the Toeplitz algebra, Cuntz algebras, and $q$-deformed spheres.

中文翻译：

---

图 C*-代数的等变维数

我们通过研究图 $C^*$-代数上的规范作用以及规范作用对有限循环子群的限制来探索最近引入的局部平凡维度。对于有限无环图和有限环的$C^*$-代数，我们刻画了这些维度的有限性，并进一步研究了许多图$C^*$-代数例子的规范作用。这些包括 Toeplitz 代数、Cuntz 代数和 $q$-变形球体。