A groupoid generalisation of Leavitt path algebras

Abstract

Let \(G\) be a locally compact, Hausdorff, etale 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 \(A(G)\).