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}
}
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… 

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References

SHOWING 1-8 OF 8 REFERENCES

Twisted $C^*$-algebras associated to finitely aligned higher-rank graphs

We introduce twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs and give a comprehensive treatment of their fundamental structural properties. We establish

The Toeplitz algebra of a Hilbert bimodule

Suppose a C*-algebra A acts by adjointable operators on a Hilbert A-module X. Pimsner constructed a C*-algebra O_X which includes, for particular choices of X, crossed products of A by Z, the Cuntz

Groupoids and C * -algebras for categories of paths

In this paper we describe a new method of defining C*-algebras from oriented combinatorial data, thereby generalizing the constructions of algebras from directed graphs, higher-rank graphs, and

Operator Algebras: Theory of C*-Algebras and von Neumann Algebras

Operators on Hilbert Space.- C*-Algebras.- Von Neumann Algebras.- Further Structure.- K-Theory and Finiteness.

Amenable groupoids

  • With a foreword by Georges Skandalis and Appendix B by E. Germain, L’Enseignement Mathématique, Geneva
  • 2000

A Groupoid Approach to C*-Algebras

A class of C * -algebras generalizing both Cuntz-Krieger algebras and crossed products by Z

  • Free probability theory
  • 1995