论文标题
反向半群和od o tale groupoid代数的简单性
Simplicity of inverse semigroup and étale groupoid algebras
论文作者
论文摘要
在本文中，我们证明，具有完全断开单位空间的étalegroupoid的代数在一个字段上具有一个简单的代数，并且仅当gropsoid最小和有效时，并且代数的唯一函数在每个开放子集中消失的唯一功能是无效的。先前关于该主题的工作要求群体类固醇在非Hausdorff案中也是拓扑主体，但我们没有。此外，我们提供了第一个示例，这些例子是具有完全断开的单位空间的最低和有效但不是拓扑主体的典型典型类别。我们的示例来自无数群体的自相似小组行动。更普遍地，我们表明，典型群体的基本代数（函数理想在每个开放式上消失的商的商）很简单，并且仅当群体固醇是最小和拓扑的，并且在代数环境中概括为$ c^*$ - 代数 - 代数 - 代数的结果。 我们工作的主要应用是提供对简单合同的反向半群代数的描述，从而回答了七十年代的Munn问题。 使用Galois Descent，我们表明，étalegroupoid和逆半群代数的简单性仅取决于该场的特征，并且可以将其从阳性特征到特征到特征$ 0 $。我们还提供了与规定的主要特征集之外的简单代数的反向半群和欧特尔类固定的示例。
In this paper, we prove that the algebra of an étale groupoid with totally disconnected unit space has a simple algebra over a field if and only if the groupoid is minimal and effective and the only function of the algebra that vanishes on every open subset is the null function. Previous work on the subject required the groupoid to be also topologically principal in the non-Hausdorff case, but we do not. Furthermore, we provide the first examples of minimal and effective but not topologically principal étale groupoids with totally disconnected unit spaces. Our examples come from self-similar group actions of uncountable groups. More generally, we show that the essential algebra of an étale groupoid (the quotient by the ideal of functions vanishing on every open set), is simple if and only if the groupoid is minimal and topologically free, generalizing to the algebraic setting a recent result for essential $C^*$-algebras. The main application of our work is to provide a description of the simple contracted inverse semigroup algebras, thereby answering a question of Munn from the seventies. Using Galois descent, we show that simplicity of étale groupoid and inverse semigroup algebras depends only on the characteristic of the field and can be lifted from positive characteristic to characteristic $0$. We also provide examples of inverse semigroups and étale groupoids with simple algebras outside of a prescribed set of prime characteristics.
