Groupoid algebras as Cuntz-Pimsner algebras

  • Adam Rennie
  • David Robertson
  • Aidan Sims

Abstract

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 $C^*_r(G_0)$.

References

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. http://dx.doi.org/10.1007/3-540-28517-2

Fowler, N. J. and Raeburn, I., The Toeplitz algebra of a Hilbert bimodule, Indiana Univ. Math. J. 48 (1999), no. 1, 155–181. http://dx.doi.org/10.1512/iumj.1999.48.1639

Katsura, T., On $C^*$-algebras associated with $C^*$-correspondences, J. Funct. Anal. 217 (2004), no. 2, 366–401. http://dx.doi.org/10.1016/j.jfa.2004.03.010

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. http://dx.doi.org/10.1090/S0002-9947-2014-06008-X

>

Published
2017-02-23
How to Cite
Rennie, A., Robertson, D., & Sims, A. (2017). Groupoid algebras as Cuntz-Pimsner algebras. MATHEMATICA SCANDINAVICA, 120(1), 115-123. https://doi.org/10.7146/math.scand.a-25507
Section
Articles