Twisted Steinberg algebras

Becky Armstrong, Lisa Orloff Clark, Kristin Courtney, Ying-Fen Lin, Kathryn McCormick and Jacqui Ramagge

Abstract

We introduce twisted Steinberg algebras, which generalise complex Steinberg algebras and are a purely algebraic notion of Renault's twisted groupoid \(C^*\)-algebras. In particular, for each ample Hausdorff groupoid \(G\) and each locally constant 2-cocycle \(\sigma\) on \(G\) taking values in the complex unit circle, we study the complex \(*\)-algebra \(A(G,\sigma)\) consisting of locally constant compactly supported functions on \(G\), with convolution and involution twisted by \(\sigma\). We also introduce a "discretised" analogue of a twist \(\Sigma\) 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 \(\Sigma\) arising from a locally constant 2-cocycle \(\sigma\) on an ample Hausdorff groupoid \(G\), we construct an associated Steinberg algebra \(A(G;\Sigma)\), and we show that it coincides with \(A(G,\sigma)\). We also prove a graded uniqueness theorem for \(A(G,\sigma)\), and under the additional hypothesis that \(G\) is effective, we prove a Cuntz–Krieger uniqueness theorem and show that simplicity of \(A(G,\sigma)\) is equivalent to minimality of \(G\).

Keywords: Steinberg algebra, topological groupoid, cohomology, graded algebra.

AMS Subject Classification: Primary 16S99; secondary (primary), 22A22 (secondary).