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 $R^{\times }$, we study the algebra $A_R(G,\sigma )$ 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 $A_R(G;\mathrm{\Sigma })$, and we show that it coincides with $A_R(G,{\sigma }^{-1})$. Given any discrete field ${\mathbb{F}}_d$, we prove a graded uniqueness theorem for $A_{{\mathbb{F}}_d}(G,\sigma )$, and under the additional hypothesis that G is effective, we prove a Cuntz–Krieger uniqueness theorem and show that simplicity of $A_{{\mathbb{F}}_d}(G,\sigma )$ is equivalent to minimality of G.

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σ上ģ的单位服用值${\mathrm{电阻}}^{\times }$, 我们学习代数 ${\mathrm{一种}}_{\mathrm{电阻}}(G,\sigma )$由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广群摹，我们表明，有在局部恒定2-余循环间的一种一一对应摹和离散曲折在摹承认连续全局部分. 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${\mathrm{一种}}_{\mathrm{电阻}}(G;\mathrm{\Sigma })$，我们证明它与 ${\mathrm{一种}}_{\mathrm{电阻}}(G,{\sigma }^{-1})$. 给定任何离散域${\mathbb{F}}_d$，我们证明了一个分级唯一性定理 ${\mathrm{一种}}_{{\mathbb{F}}_d}(G,\sigma )$，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${\mathrm{一种}}_{{\mathbb{F}}_d}(G,\sigma )$等价于G 的极小性。