通过投影对Roe代数与准局部代数的K-理论比较
$K$-Theoretic Comparison of Roe and Quasi-Local Algebras via Projections
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中文总结 AI 辅助
本文针对稀疏度量空间，证明了关于Roe代数与准局部代数K-理论的三项结果，给出了二者K-理论不同的首个否定实例。
中文摘要 AI 辅助
Roe代数与准局部代数是与度量空间相关的C*-代数，在高阶指标理论与算子代数中具有重要作用。核心问题是这两个代数及其K-理论是否相同，该问题在近年受到广泛关注，并在几何、拓扑与分析领域衍生出诸多应用。本文聚焦于稀疏度量空间X（即X为有限子空间X_n的不交并，且当n≠m时，X_n与X_m的距离趋于正无穷）以及块对角投影P（即P为强算子拓扑下对P_n的求和，其中P_n属于ℓ²(X_n)上的有界线性算子代数），证明了三项主要结果：(1) 若各P_n的秩一致有界，则P为准局部的当且仅当它属于一致Roe代数；(2) 一般情况下，存在属于准局部但不属于一致Roe代数的幽灵投影；(3) 若{X_n}为满足额外围长条件的扩张图，则一致Roe代数到一致准局部代数的嵌入在其K₀群上不诱导同构，这是已知的首个关于二者K-理论的否定结果。
英文摘要
A central question in higher index theory and operator algebras is whether the Roe algebra and the quasi-local algebra associated with a metric space of bounded geometry coincide, or at least have the same $K$-theory. In this paper, we focus on a \emph{sparse} metric space $X$. We prove the following three main results: (1) For a block-diagonal operator $T$ with uniformly bounded block-rank, $T$ is quasi-local if and only if it is in the Roe algebra. (2) In general, we discover a ghost block-diagonal projection which is quasi-local but not in the Roe algebra. (3) For a sequence of expander graphs with sufficiently large girth, the inclusion of the uniform Roe algebra into the uniform quasi-local algebra induces a \emph{non-surjective} map on their $K_0$-groups. This yields the first known $K$-theoretic distinction between the uniform Roe algebra and the uniform quasi-local algebra.
发表机构
- Centre for Mathematical Sciences, Lund University（隆德大学数学中心）
- School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）
- College of Data Science, Jiaxing University（嘉兴大学数据科学学院）
机构由 AI 辅助整理，请以论文原文为准。
