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A groupoid generalisation of Leavitt path algebras

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Abstract

Let \(G\) be a locally compact, Hausdorff, étale groupoid whose unit space is totally disconnected. We show that the collection \(A(G)\) of locally-constant, compactly supported complex-valued functions on \(G\) is a dense \(*\)-subalgebra of \(C_c(G)\) and that it is universal for algebraic representations of the collection of compact open bisections of \(G\). We also show that if \(G\) is the groupoid associated to a row-finite graph or \(k\)-graph with no sources, then \(A(G)\) is isomorphic to the associated Leavitt path algebra or Kumjian–Pask algebra. We prove versions of the Cuntz–Krieger and graded uniqueness theorems for \(A(G)\).

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Notes

  1. We would like to thank Steinberg who brought this to our attention after reading an earlier version of this paper.

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Acknowledgments

This work was partially supported by a grant from the Simons Foundation (#210035 to Mark Tomforde).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Otago, PO Box 56, Dunedin , 9054, New Zealand

    Lisa Orloff Clark

  2. Department of Mathematics, 14 MacLean Hall, Iowa City, IA , 52242-1419, USA

    Cynthia Farthing

  3. School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW , 2518, Australia

    Aidan Sims

  4. Department of Mathematics, University of Houston, Houston, TX , 77204-3008, USA

    Mark Tomforde

Authors
  1. Lisa Orloff Clark
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  2. Cynthia Farthing
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  3. Aidan Sims
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  4. Mark Tomforde
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Correspondence to Lisa Orloff Clark.

Additional information

Communicated by Benjamin Steinberg.

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Clark, L.O., Farthing, C., Sims, A. et al. A groupoid generalisation of Leavitt path algebras. Semigroup Forum 89, 501–517 (2014). https://doi.org/10.1007/s00233-014-9594-z

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