发表机构
Universidade Federal de Santa Catarina; Research Institute of Mathematics, Seoul National University（圣卡塔琳娜联邦大学; 首尔大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过建立局部紧Hausdorff étale群胚的拉回定理，结合边界路径分解定理，统一解释了图、相对图和拓扑图$C^*$-代数的拉回现象。
AI 中文摘要
我们建立了局部紧Hausdorff étale群胚的$C^*$-代数的拉回定理。该定理表明，当单位空间被两个闭不变子集覆盖，且其补集为不相交的开不变子集时，相应的群胚$C^*$-代数在$\mathbb T$-$C^*$-代数及$\mathbb T$-等变$*$-同态的范畴中构成一个拉回图，其中规范作用由$\mathbb Z$-值余循环及其限制诱导。随后，我们证明了图、相对图和拓扑图的一系列边界路径分解定理。我们表明，图的可容许分解及其在相对和拓扑情形下的类似物，诱导边界路径空间的相应分解。将这些分解定理与群胚拉回定理相结合，我们恢复了图$C^*$-代数、相对图$C^*$-代数和拓扑图$C^*$-代数先前已知的拉回定理。因此，这些拉回现象可由单一的群胚理论机制解释。
英文摘要
We establish a pullback theorem for \(C^*\)-algebras of locally compact Hausdorff étale groupoids. The theorem shows that, when the unit space is covered by two closed invariant subsets whose complements are disjoint open invariant subsets, the corresponding groupoid \(C^*\)-algebras form a pullback diagram in the category of \(\mathbb T\)-\(C^*\)-algebras and \(\mathbb T\)-equivariant \(*\)-homomorphisms, for the gauge actions induced by a \(\mathbb Z\)-valued cocycle and its restrictions. We then prove a collection of boundary-path decomposition theorems for graphs, relative graphs, and topological graphs. We show that admissible decompositions of graphs, together with their analogues in the relative and topological settings, induce corresponding decompositions of boundary path spaces. Combining these decomposition theorems with the groupoid pullback theorem, we recover previously known pullback theorems for graph \(C^*\)-algebras, relative graph \(C^*\)-algebras, and topological graph \(C^*\)-algebras. Thus these pullback phenomena are explained by a single groupoid-theoretic mechanism.
Comments26 pages
