Twisted Steinberg algebras

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,σ−1). 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.