标题： $L^p$粗 Baum-Connes 猜想通过$C_{0}$粗几何 显示英文标题

标题： $L^p$ coarse Baum-Connes conjecture via $C_{0}$ coarse geometry

摘要： 在本文中，我们通过$C_{0}$粗结构研究$p\in [1,\infty)$的$L^{p}$粗 Baum-Connes 猜想，这是度量空间上有界粗结构的改进。 我们证明了在配备了一致球面度量的有限维单纯复形上，$L^{p}$粗 Baum-Connes 猜想的$C_{0}$版本成立。 利用这一结果，我们构造了$L^{p}$粗 Baum-Connes 猜想的障碍群。 作为应用，我们表明在有限渐近维数假设下，障碍群消失，从而为此种情况提供了$L^{p}$粗 Baum-Connes 猜想的新证明。 显示英文摘要

摘要： In this paper, we investigate the $L^{p}$ coarse Baum-Connes conjecture for $p\in [1,\infty)$ via $C_{0}$ coarse structure, which is a refinement of the bounded coarse structure on a metric space. We prove that the $C_{0}$ version of the $L^{p}$ coarse Baum-Connes conjecture holds for a finite-dimensional simplicial complex equipped with a uniform spherical metric. Using this result, we construct an obstruction group for the $L^{p}$ coarse Baum-Connes conjecture. As an application, we show that the obstruction group vanishes under the assumption of finite asymptotic dimension, thereby providing a new proof of the $L^{p}$ coarse Baum-Connes conjecture in this case.