# 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}
}
• 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 $C… • Mathematics • 2018 Given a$\mathbb{Z}$-graded ring$A$and a subring$R\subseteq A$, it is natural to ask whether$A$can be realised as the Cuntz-Pimsner ring of some$R\$-system. In this paper, we derive sufficient
• Mathematics
Journal of Operator Theory
• 2021
Suppose G is a second-countable locally compact Hausdorff \'{e}tale groupoid, G is a discrete group containing a unital subsemigroup P, and c:G→G is a continuous cocycle. We derive conditions on the
• Mathematics
• 2021
We consider a locally compact Hausdorff groupoid G which is graded over a discrete group. Then the fibre over the identity is an open and closed subgroupoid Ge. We show that both the full and reduced
• Mathematics
• 2017
We prove that if A is a \sigma-unital exact C*-algebra of real rank zero, then every state on K_0(A) is induced by a 2-quasitrace on A. This yields a generalisation of Rainone's work on pure
• Mathematics
• 2016
We define and study fibrations of topological groupoids. We interpret a groupoid fibration L->H with fibre G as an action of H on G by groupoid equivalences. Our main result shows that a crossed
• Mathematics
Annales Henri Poincaré
• 2019
We examine the non-commutative index theory associated with the dynamics of a Delone set and the corresponding transversal groupoid. Our main motivation comes from the application to topological
• Mathematics
Annales Henri Poincaré
• 2019
We examine the non-commutative index theory associated with the dynamics of a Delone set and the corresponding transversal groupoid. Our main motivation comes from the application to topological

## References

SHOWING 1-8 OF 8 REFERENCES

• Mathematics
Documenta Mathematica
• 2014
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
• Mathematics
• 1998
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
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
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 class of C * -algebras generalizing both Cuntz-Krieger algebras and crossed products by Z

• Free probability theory
• 1995