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

@article{Sims2013TwistedA, title={Twisted \$C^*\$-algebras associated to finitely aligned higher-rank graphs}, author={Aidan Sims and Benjamin Whitehead and Michael F. Whittaker}, journal={Documenta Mathematica}, year={2013} }

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 versions of the usual uniqueness theorems and the classification of gauge-invariant ideals. We show that all twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs are nuclear and satisfy the UCT, and that for twists that lift to real-valued cocycles, the $K$-theory of…

## 30 Citations

### Product-system models for twisted C⁎-algebras of topological higher-rank graphs

- MathematicsJournal of Mathematical Analysis and Applications
- 2018

### Simplicity of twisted C*-algebras of higher-rank graphs and crossed products by quasifree actions

- Mathematics
- 2014

We characterise simplicity of twisted C*-algebras of row-finite k-graphs with no sources. We show that each 2-cocycle on a cofinal k-graph determines a canonical second-cohomology class for the…

### Twisted k-Graph Algebras Associated to Bratteli Diagrams

- Mathematics
- 2014

Given a system of coverings of k-graphs, we show that the second cohomology of the resulting (k + 1)-graph is isomorphic to that of any one of the k-graphs in the system, and compute the semifinite…

### Twisted k-Graph Algebras Associated to Bratteli Diagrams

- MathematicsIntegral Equations and Operator Theory
- 2015

Given a system of coverings of k-graphs, we show that the second cohomology of the resulting (k + 1)-graph is isomorphic to that of any one of the k-graphs in the system, and compute the semifinite…

### Efficient Presentations of Relative Cuntz-Krieger Algebras

- MathematicsAnalysis Mathematica
- 2021

In this article, we present a new method to study relative Cuntz-Krieger algebras for higher-rank graphs. We only work with edges rather than paths of arbitrary degrees. We then use this method to…

### Efficient Presentations of Relative Cuntz-Krieger Algebras

- MathematicsAnalysis Mathematica
- 2021

In this article, we present a new method to study relative Cuntz-Krieger algebras for higher-rank graphs. We only work with edges rather than paths of arbitrary degrees. We then use this method to…

### GRAPHS AND CROSSED PRODUCTS BY QUASIFREE ACTIONS

- Mathematics
- 2014

We characterise simplicity of twisted C � -algebras of row-finite k-graphs with no sources. We show that each 2-cocycle on a cofinal k-graph determines a canonical second-cohomology class for the…

### Simplicity of twisted C*-algebras of Deaconu--Renault groupoids

- Mathematics
- 2021

We consider Deaconu–Renault groupoids associated to actions of finite-rank free abelian monoids by local homeomorphisms of locally compact Hausdorff spaces. We study simplicity of the twisted…

### KMS states on the C*-algebras of Fell bundles over groupoids

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2019

Abstract We consider fibrewise singly generated Fell bundles over étale groupoids. Given a continuous real-valued 1-cocycle on the groupoid, there is a natural dynamics on the cross-sectional algebra…

## References

SHOWING 1-10 OF 32 REFERENCES

### Twisted relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs

- Mathematics
- 2013

To each finitely aligned higher-rank graph $\Lambda$ and each $\mathbb{T}$-valued 2-cocycle on $\Lambda$, we associate a family of twisted relative Cuntz-Krieger algebras. We show that each of these…

### HOMOLOGY FOR HIGHER-RANK GRAPHS AND TWISTED

- Mathematics
- 2011

We introduce a homology theory for k-graphs and explore its fundamental properties. We establish connections with algebraic topology by showing that the homol- ogy of a k-graph coincides with the…

### Co-universal C*-algebras associated to generalised graphs

- Mathematics
- 2010

We introduce P-graphs, which are generalisations of directed graphs in which paths have a degree in a semigroup P rather than a length in ℕ. We focus on semigroups P arising as part of a…

### Relative Cuntz-Krieger algebras of finitely aligned higher-rank graphs

- Mathematics
- 2003

We define the relative Cuntz-Krieger algebras associated to finitely aligned higher-rank graphs. We prove versions of the gauge-invariant unique- ness theorem and the Cuntz-Krieger uniqueness theorem…

### Gauge-Invariant Ideals in the C*-Algebras of Finitely Aligned Higher-Rank Graphs

- MathematicsCanadian Journal of Mathematics
- 2006

Abstract We produce a complete description of the lattice of gauge-invariant ideals in ${{C}^{*}}(\Lambda )$ for a finitely aligned $k$ -graph $\Lambda $ . We provide a condition on $\Lambda $ under…

### Product systems of graphs and the Toeplitz algebras of higher-rank graphs | NOVA. The University of Newcastle's Digital Repository

- Mathematics
- 2003

There has recently been much interest in the C � -algebras of directed graphs. Here we consider product systems E of directed graphs over semigroups and associated C � -algebras C � (E) and T C � (E)…

### The primitive ideals of the Cuntz–Krieger algebra of a row-finite higher-rank graph with no sources ☆

- Mathematics
- 2014

### THE C -ALGEBRAS OF ROW-FINITE GRAPHS

- Mathematics
- 2000

NSKI Abstract. We prove versions of the fundamental theorems about Cuntz-Krieger algebras for the C -algebras of row-finite graphs: directed graphs in which each vertex emits at most finitely many…