Twisted Steinberg algebras

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:226

Article

Armstrong, Becky¹  Clark, Lisa Orloff²  Courtney, Kristin³  Lin, Ying-Fen⁴  McCormick, Kathryn⁵  Ramagge, Jacqui⁶ 

[1] Univ Wollongong, Sch Math & Appl Stat, Northfields Ave, Wollongong, NSW 2522, Australia

[2] Victoria Univ Wellington, Sch Math & Stat, POB 600, Wellington 6140, New Zealand

[3] WWU Munster, Math Inst, Einsteinstr 62, D-48149 Munster, Germany

[4] Queens Univ Belfast, Math Sci Res Ctr, Belfast BT7 1NN, Antrim, North Ireland

[5] Calif State Univ Long Beach, Dept Math & Stat, Long Beach, CA 90840 USA

[6] Univ Durham, Fac Sci, Durham DH1 3LE, England

关键词: Steinberg algebra;    Topological groupoid;    Cohomology;    Graded algebra;   

DOI  :  10.1016/j.jpaa.2021.106853

来源: Elsevier

【 摘 要 】

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-co cycle sigma on G taking values in the units R-x , we study the algebra A(R)(G, sigma) consisting of locally constant compactly supported R-valued functions on G , with convolution and involution twisted by sigma. We also introduce a discretised analogue of a twist sigma over a Hausdorff etale groupoid G , and we show that there is a one-to-one correspondence between locally constant 2-co cycles on G and discrete twists over G admitting a continuous global section. Given a discrete twist sigma arising from a locally constant 2-co cycle sigma on an ample Hausdorff groupoid G, we construct an associated twisted Steinberg algebra A(R)(G; Sigma), and we show that it coincides with A(R)(G,sigma(-1)). Given any discrete field F-d, we prove a graded uniqueness theorem for A(Fd) (G,sigma), and under the additional hypothesis that G is effective, we prove a Cuntz-Krieger uniqueness theorem and show that simplicity of A(Fd) (G, sigma) is equivalent to minimality of G. (C) 2021 Elsevier B.V. All rights reserved.

【 授权许可】

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