A groupoid generalization of Leavitt path 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).

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• Reconstruction of twisted Steinberg algebras
• A Generalised uniqueness theorem and the graded ideal structure of Steinberg algebras
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• Simplicity of algebras via epsilon-strong systems
• Equivalent groupoids have Morita equivalent Steinberg algebras
• Simplicity of algebras associated to étale groupoids
• $\mathbb{Z}$-graded rings as Cuntz-Pimsner rings