Semigroup Forum

, Volume 89, Issue 3, pp 501–517 | Cite as

A groupoid generalisation of Leavitt path algebras

  • Lisa Orloff Clark
  • Cynthia Farthing
  • Aidan Sims
  • Mark Tomforde


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 \(A(G)\).


Topological groupoids Leavitt algebra Groupoid algebra  Graded algebra 



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


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Lisa Orloff Clark
    • 1
  • Cynthia Farthing
    • 2
  • Aidan Sims
    • 3
  • Mark Tomforde
    • 4
  1. 1.Department of Mathematics and StatisticsUniversity of OtagoDunedin New Zealand
  2. 2.Department of MathematicsIowa CityUSA
  3. 3.School of Mathematics and Applied StatisticsUniversity of WollongongWollongongAustralia
  4. 4.Department of MathematicsUniversity of HoustonHoustonUSA

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