A groupoid generalization of Leavitt path algebras

TL;DR

This paper generalizes Leavitt path algebras using groupoid theory, establishing a universal algebraic framework for locally-constant functions on certain groupoids and connecting it to known algebraic structures like Leavitt path algebras.

Contribution

It introduces a groupoid-based generalization of Leavitt path algebras, extending their algebraic and analytical properties to broader classes of groupoids.

Findings

A(G) is a dense *-subalgebra of C_c(G).

A(G) is universal for algebraic representations of compact open bisections.

For specific groupoids, A(G) is isomorphic to known Leavitt path or Kumjian-Pask algebras.

Abstract

Let G be a locally compact, Hausdorff groupoid in which s is a local homeomorphism and the unit space is totally disconnected. Assume there is a continuous cocycle c from G into a discrete group $\Gamma$. We show that the collection A(G) of locally-constant, compactly supported 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).