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 $C^{\ast }$-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 $\ast$-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 $\mathrm{\Sigma }$ 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 $\mathrm{\Sigma }$ arising from a locally constant 2-cocycle $\sigma$ on an ample Hausdorff groupoid $G$, we construct an associated Steinberg algebra $A(G;\mathrm{\Sigma })$, 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).