Twisted Steinberg algebras

We introduce twisted Steinberg algebras over a commutative unital ring $R$. These generalise and are purely algebraic analogue of Renault's groupoid C*-algebras. In particular, for each ample Hausdorff $G$ locally constant $2$-cocycle $\sigma$ on taking values in the units $R^\times$, we study algebra $A_R(G,\sigma)$ consisting compactly supported $R$-valued functions $G$, with convolution involution "twisted" by $\sigma$. also "discretised" twist $\Sigma$ \'etale show that there is one-to-one correspondence between $2$-cocycles discrete twists admitting continuous global section. Given arising from an construct associated $A_R(G;\Sigma)$, it coincides $A_R(G,\sigma^{-1})$. any field $\mathbb{F}_d$, prove graded uniqueness theorem $A_{\mathbb{F}_d}(G,\sigma)$, under additional hypothesis effective, Cuntz--Krieger simplicity $A_{\mathbb{F}_d}(G,\sigma)$ equivalent to minimality $G$.