• DOI:10.7146/MATH.SCAND.A-25507
• Corpus ID: 115549060

Groupoid algebras as Cuntz-Pimsner algebras

@article{Rennie2014GroupoidAA,
  title={Groupoid algebras as Cuntz-Pimsner algebras},
  author={Adam Graham Rennie and David I. Robertson and Aidan Sims},
  journal={arXiv: Operator Algebras},
  year={2014}
}

• A. Rennie, D. Robertson, A. Sims
• Published 28 February 2014
• Mathematics
• arXiv: Operator Algebras

We show that if $G$ is a second countable locally compact Hausdorff \'etale groupoid carrying a suitable cocycle $c:G\to\mathbb{Z}$, then the reduced $C^*$-algebra of $G$ can be realised naturally as the Cuntz-Pimsner algebra of a correspondence over the reduced $C^*$-algebra of the kernel $G_0$ of $c$. If the full and reduced $C^*$-algebras of $G_0$ coincide, we deduce that the full and reduced $C^*$-algebras of $G$ coincide. We obtain a six-term exact sequence describing the $K$-theory of $C^*_r(G)$ in terms of that of $C^*_r(G_0)$.