论文标题
of for类群的准局部性
Quasi-locality for étale groupoids
论文作者
论文摘要
令$ \ MATHCAL {G} $为本地紧凑的Étalegroupoid，$ \ Mathscr {l}（l^2（\ Mathcal {G}））$为$ C^*$ - 在Hilbert $ c^*$ c^*$ - module $ l^2（Mathcalcal c}）上可邻接的操作员的代数 - 在本文中，我们在$ \ mathscr {l}（l^2（\ Mathcal {g}}））$中发现了一个称为运算符的Quasi-locality的概念，概括了ROE引入的度量空间案例。我们的主要结果表明，当$ \ Mathcal {g} $还为$σ$ -compact andable时，是$ \ Mathscr {l}（l^2（\ Mathcal {g}）中的均等运算符时准本地。这提供了一种实用的方法来描述$ c^*_ r（\ Mathcal {g}）$使用粗几何形状中的元素。我们的主要工具是$ \ mathscr {l}（l^2（\ Mathcal {g}））$的操作员的描述，其切片与计算机断层扫描具有相同的理念。作为应用程序，我们恢复了Špakula和公制空间案例中的第二名作者的结果，并推导了减少跨产品和统一的ROE代数的新特征。
Let $\mathcal{G}$ be a locally compact étale groupoid and $\mathscr{L}(L^2(\mathcal{G}))$ be the $C^*$-algebra of adjointable operators on the Hilbert $C^*$-module $L^2(\mathcal{G})$. In this paper, we discover a notion called quasi-locality for operators in $\mathscr{L}(L^2(\mathcal{G}))$, generalising the metric space case introduced by Roe. Our main result shows that when $\mathcal{G}$ is additionally $σ$-compact and amenable, an equivariant operator in $\mathscr{L}(L^2(\mathcal{G}))$ belongs to the reduced groupoid $C^*$-algebra $C^*_r(\mathcal{G})$ if and only if it is quasi-local. This provides a practical approach to describe elements in $C^*_r(\mathcal{G})$ using coarse geometry. Our main tool is a description for operators in $\mathscr{L}(L^2(\mathcal{G}))$ via their slices with the same philosophy to the computer tomography. As applications, we recover a result by Špakula and the second-named author in the metric space case, and deduce new characterisations for reduced crossed products and uniform Roe algebras for groupoids.
