Deaconu-Renault 群胚中的轨道破缺及 UCT Kirchberg 代数的模型
Orbit-breaking in Deaconu-Renault groupoids and models for UCT Kirchberg algebras
浏览论文内容
中文总结 AI 辅助
本文将轨道破缺推广到满射局部同胚，构造出开子群胚，其C*-代数为UCT Kirchberg代数，并给出到O_2的嵌入，实现所有稳定UCT Kirchberg代数及K-群同态诱导的*-同态。
中文摘要 AI 辅助
我们将轨道破缺的概念从同胚情形推广到满射局部同胚情形。该方法在满射局部同胚关联的 Deaconu--Renault 群胚中构造开子群胚。当映射满足自然假设（包括局部扩张性）时，所得群胚 C*-代数为单的 UCT Kirchberg 代数。我们构造了一类特殊的 Deaconu--Renault 系统，每个系统都模拟 Cuntz 代数 O_2。对此类系统进行轨道破缺得到的 C*-代数具有到 O_2 的显式嵌入。我们还引入了一种在移除不动点后进行轨道破缺的方法。利用该方法，我们可以实现所有稳定的 UCT Kirchberg 代数，且每个代数都带有到 O_2⊗K 的嵌入。此外，我们证明，给定可数分次阿贝尔群之间的同态，存在由群胚包含诱导的轨道破缺代数之间的 *-同态。特别地，我们从 K-群之间的任意群同态获得稳定 UCT Kirchberg 代数之间的显式 *-同态。
英文摘要
We generalize the concept of orbit-breaking from the case of a homeomorphism to the case of a surjective local homeomorphism. This method creates open subgroupoids of the Deaconu--Renualt groupoid associated to the surjective local homeomorphism. When the map satisfies natural assumptions (including being locally expanding) the resulting groupoid $\mathrm{C}^*$-algebras are unital UCT Kirchberg algebras. A specific class of Deaconu--Renault systems are constructed, each modelling the Cuntz algebra $\mathcal{O}_2$. The $\mathrm{C}^*$-algebras obtained from orbit-breaking for such systems come with explicit embeddings into $\mathcal{O}_2$. We also introduce a method for orbit-breaking after removing a fixed point. Using this method, we can realize all stable UCT Kirchberg algebras with each coming with an embeddings into $\mathcal O_2\otimes\mathcal K$. Moreover, we show that, given a homomorphism between countable graded abelian groups, there is a $^*$-homomorphism between orbit-breaking algebras obtained from a groupoid inclusion. In particular, we obtain explicit $^*$-homomorphisms between stable UCT Kirchberg algebras from any group homomorphism between their $K$-groups.
发表机构
- University of Colorado Boulder（科罗拉多大学博尔德分校）
- University of Victoria（维多利亚大学）
- Institute of Mathematics, Czech Academy of Sciences（捷克科学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
