A groupoid generalisation of Leavitt path algebras

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 Cc(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).