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 A _R(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 A _R(G;Σ), and we show that it coincides with A _R(G,σ ⁻¹). Given any discrete field F _d, we prove a graded uniqueness theorem for A _{F d} (G,σ), and under the additional hypothesis that G is effective, we prove a Cuntz–Krieger uniqueness theorem and show that simplicity of A _{F d} (G,σ) is equivalent to minimality of G.