AI 中文总结
该研究针对平展群胚上的 Fell 丛的 C*代数，建立其基态空间与对应边界群胚 C*代数态空间的仿射同胚，探讨基态与 KMS_∞ 态的关系，并将结果应用于两类扭曲 C*代数以得到基态的显式刻画。
AI 中文摘要
设 p: 𝒜→G 是局部紧豪斯多夫第二可数平展群胚 G 上的 Fell 丛，C*(G;𝒜) 是对应的 Fell 丛 C*代数。设 σ^c 是由实值 1-上循环 c 诱导的 C*(G;𝒜) 上的实动力学，我们建立了 C*(G;𝒜) 的 σ^c-基态空间与 C*(G(Z);𝒜|_{G(Z)}) 的态空间之间的仿射同胚，其中 Z 是 c 的边界集，G(Z) 是边界群胚。特别地，C*(G;𝒜) 存在 σ^c-基态当且仅当 Z≠∅。我们进一步研究了基态与 KMS_∞ 态的关系，并当上循环是局部常值时，给出了边界 C*代数 C*(G(Z);𝒜|_{G(Z)}) 的结晶解释。最后，我们将结果应用于几类例子，包括 Deaconu–Renault 群胚的扭曲 C*代数和高阶图的扭曲 C*代数，得到了它们基态的显式刻画。
英文摘要
Let \(p\colon \mathcal{A}\to G\) be a Fell bundle over a locally compact Hausdorff second countable étale groupoid \(G\) and let \(\mathrm{C}^*(G; \mathcal{A})\) be the associated Fell bundle \(\mathrm{C}^*\)-algebra. Suppose \(σ^c\) is the real dynamics on \(\mathrm{C}^*(G;\mathcal{A})\) induced by a real-valued \(1\)-cocycle \(c\). We establish an affine homeomorphism between the \(σ^c\)-ground state space of \(\mathrm{C}^*(G;\mathcal{A})\) and the state space of \(\mathrm{C}^*(G(Z); \mathcal{A}|_{G(Z)})\), where \(Z\) is the boundary set of \(c\) and \(G(Z)\) is the boundary groupoid. In particular, \(\mathrm{C}^*(G; \mathcal{A})\) admits \(σ^c\)-ground states if and only if \(Z\neq \emptyset\). We further investigate the relation between ground states and KMS\(_{\infty}\) states and give a crystallization interpretation of the boundary \(\mathrm{C}^*\)-algebra \(\mathrm{C}^*(G(Z); \mathcal{A}|_{G(Z)})\) when the cocycle is locally constant. Finally, we apply our results to several classes of examples, including twisted \(\mathrm{C}^*\)-algebras of Deaconu--Renault groupoids and twisted higher-rank graph \(\mathrm{C}^*\)-algebras, obtaining explicit descriptions of their ground states.
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