Open Access Open Access  Restricted Access Subscription Access

Groupoid algebras as Cuntz-Pimsner algebras

Adam Rennie, David Robertson, Aidan Sims


We show that if $G$ is a second countable locally compact Hausdorff étale groupoid carrying a suitable cocycle $c\colon G\to\mathbb{Z}$, 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 $C^*_r(G_0)$.

Full Text:



Anantharaman-Delaroche, C. and Renault, J., Amenable groupoids, Monographies de L'Enseignement Mathématique, vol. 36, L'Enseignement Mathématique, Geneva, 2000.

Blackadar, B., Operator algebras: Theory of $C^*$-algebras and von Neumann algebras, Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006.

Fowler, N. J. and Raeburn, I., The Toeplitz algebra of a Hilbert bimodule, Indiana Univ. Math. J. 48 (1999), no. 1, 155–181.

Katsura, T., On $C^*$-algebras associated with $C^*$-correspondences, J. Funct. Anal. 217 (2004), no. 2, 366–401.

Pimsner, M. V., A class of $C^*$-algebras generalizing both Cuntz-Krieger algebras and crossed products by $bf Z$, Free probability theory (Waterloo, ON, 1995), Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 189--212.

Renault, J., A groupoid approach to $C^ast$-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980.

Sims, A., Whitehead, B., and Whittaker, M. F., Twisted $C^*$-algebras associated to finitely aligned higher-rank graphs, Doc. Math. 19 (2014), 831–866.

Spielberg, J., Groupoids and $C^*$-algebras for categories of paths, Trans. Amer. Math. Soc. 366 (2014), no. 11, 5771–5819.



  • There are currently no refbacks.
This website uses cookies to allow us to see how the site is used. The cookies cannot identify you or any content at your own computer.

ISSN 0025-5521 (print) ISSN 1903-1807 (online)

Hosted by the Royal Danish Library