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.

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

This paper is available as a pdf (444kB) file. It is also on the arXiv: arxiv.org/abs/1910.13005v1.

 Thursday, October 31, 2019