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Twisted Steinberg algebras
Journal of Pure and Applied Algebra ( IF 0.7 ) Pub Date : 2021-07-13 , DOI: 10.1016/j.jpaa.2021.106853
Becky Armstrong 1 , Lisa Orloff Clark 2 , Kristin Courtney 3 , Ying-Fen Lin 4 , Kathryn McCormick 5 , Jacqui Ramagge 6
Affiliation  

We introduce twisted Steinberg algebras over a commutative unital ring R. These generalise Steinberg algebras and are a purely algebraic analogue of Renault's twisted groupoid C*-algebras. In particular, for each ample Hausdorff groupoid G and each locally constant 2-cocycle σ on G taking values in the units R×, we study the algebra AR(G,σ) consisting of locally constant compactly supported R-valued functions on G, with convolution and involution “twisted” by σ. We also introduce a “discretised” analogue of a twist Σ over a Hausdorff étale groupoid G, and we show that there is a one-to-one correspondence between locally constant 2-cocycles on G and discrete twists over G admitting a continuous global section. Given a discrete twist Σ arising from a locally constant 2-cocycle σ on an ample Hausdorff groupoid G, we construct an associated twisted Steinberg algebra AR(G;Σ), and we show that it coincides with AR(G,σ1). Given any discrete field Fd, we prove a graded uniqueness theorem for AFd(G,σ), and under the additional hypothesis that G is effective, we prove a Cuntz–Krieger uniqueness theorem and show that simplicity of AFd(G,σ) is equivalent to minimality of G.



中文翻译:

扭曲的斯坦伯格代数

我们在交换单位环R 上引入了扭曲的 Steinberg 代数。这些概括了 Steinberg 代数,并且是雷诺扭曲群群 C* 代数的纯代数模拟。特别地,对于每一个充足的Hausdorff广群ģ并且每个局部恒定2-闭链σģ的单位服用值电阻×, 我们学习代数 一种电阻(G,σ)G上的局部常数紧凑支持R值函数组成,卷积和对合由σ “扭曲”。我们还引入了一个扭曲的“离散化”模拟Σ在豪斯多夫étale广群,我们表明,有在局部恒定2-余循环间的一种一一对应和离散曲折在承认连续全局部分. 给定一个由局部常数 2-cocycle σ在充足的 Hausdorff groupoid G上产生的离散扭曲 Σ ,我们构建了一个相关的扭曲 Steinberg 代数一种电阻(G;Σ),我们证明它与 一种电阻(G,σ-1). 给定任何离散域Fd,我们证明了一个分级唯一性定理 一种Fd(G,σ),并且在G有效的附加假设下,我们证明了 Cuntz-Krieger 唯一性定理并证明了一种Fd(G,σ)等价于G 的极小性

更新日期:2021-07-22
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