Twisted Steinberg algebras

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

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.
Original languageEnglish
Article number106853
Number of pages31
JournalJournal of Pure and Applied Algebra
Volume226
Issue number3
Early online date13 Jul 2021
DOIs
Publication statusEarly online date - 13 Jul 2021

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