发表机构
University College Dublin; University of Münster; University of Oxford; University of Ottawa（都柏林大学学院; 明斯特大学; 牛津大学; 渥太华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明简单可分纯核 C*-代数的 Z-稳定性，结合 Thiel 工作完成 Toms-Winter 猜想，并通过 II_1 因子迹完备代数框架给出正则性刻画及 K-稳定性等应用。
AI 中文摘要
我们证明了所有简单、可分、纯核 C*-代数的 Z-稳定性。将此与 Thiel 最近的工作相结合，完成了 Toms-Winter 猜想的证明。我们通过 II_1 因子迹完备 C*-代数的框架来实现这一点。我们用多种术语刻画这些代数的正则性。特别地，它们是正则的当且仅当它们具有实秩零，且投影的 Murray-von Neumann 等价类对应于从指定迹到 [0,1] 的连续仿射函数。作为我们对迹完备代数的超幂分析的一个应用，我们证明了拟迹为迹的简单纯 C*-代数是 K-稳定的。在 amenable 情形下，我们给出了一个 Popa 风格的局部量子化结果，针对 II_1 因子 amenable 迹完备 C*-代数，该结果从正则性条件获得超有限性。我们的 Z-稳定性结果由此连同 Matui 和 Sato 的一个定理得出。在 Bauer 迹单纯形情形中，一个关键引理的初始版本由 ChatGPT Astra 获得。
英文摘要
We prove Z-stability for all simple, separable, pure, nuclear C*-algebras. Combining this with very recent work of Thiel completes the proof of the Toms-Winter conjecture. We do this through the framework of II_1 factorial tracially complete C*-algebras. We characterise regularity for these algebras in various terms. In particular they are regular if and only if they have real rank zero and the Murray-von Neumann equivalence classes of projections correspond to continuous affine functions from the designated traces to [0,1]. As an application of our analysis of ultrapowers of tracial completions, we show simple pure C*-algebras whose quasitraces are traces are K-stable. In the amenable setting we give a local quantisation result, a la Popa, for II_1 factorial amenable tracially complete C*-algebras, which obtains hyperfiniteness from regularity conditions. Our Z-stability results follow from this together with a theorem of Matui and Sato. An initial version of an essential lemma in the Bauer trace simplex case was obtained from ChatGPT Astra.
Comments70 pages
