# Simplicity of C*‐algebras associated to higher‐rank graphs

@article{Robertson2006SimplicityOC, title={Simplicity of C*‐algebras associated to higher‐rank graphs}, author={David I. Robertson and Aidan Sims}, journal={Bulletin of the London Mathematical Society}, year={2006}, volume={39} }

We prove that if Λ is a row‐finite k‐graph with no sources, then the associated C*‐algebra is simple if and only if Λ is cofinal and satisfies Kumjian and Pask's aperiodicity condition, known as Condition (A). We prove that the aperiodicity condition is equivalent to a suitably modified version of Robertson and Steger's original nonperiodicity condition (H3), which in particular involves only finite paths. We also characterise both cofinality and aperiodicity of Λ in terms of ideals in C*(Λ).

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## References

SHOWING 1-10 OF 13 REFERENCES

### * -algebras of Finitely Aligned Higher-rank Graphs

- Mathematics
- 2003

We generalise the theory of Cuntz-Krieger families and graph algebras to the class of finitely aligned k-graphs. This class contains in particular all row-finite k-graphs. The Cuntz-Krieger relations…

### HIGHER-RANK GRAPHS AND THEIR $C^*$-ALGEBRAS

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2003

Abstract We consider the higher-rank graphs introduced by Kumjian and Pask as models for higher-rank Cuntz–Krieger algebras. We describe a variant of the Cuntz–Krieger relations which applies to…

### On higher rank graph C ∗ -algebras

- Mathematics
- 2000

Given a row-finite k-graph Λ with no sources we investigate the K-theory of the higher rank graph C *-algebra, C * (Λ). When k = 2 we are able to give explicit formulae to calculate the K-groups of C…

### Affine buildings, tiling systems and higher rank Cuntz-Krieger algebras

- Mathematics
- 1999

To an $r$-dimensional subshift of finite type satisfying certain special properties we associate a $C^*$-algebra $\cA$. This algebra is a higher rank version of a Cuntz-Krieger algebra. In…

### 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…

### CUNTZ-KRIEGER ALGEBRAS OF DIRECTED GRAPHS

- Mathematics
- 1998

We associate to each row-nite directed graph E a universal Cuntz-Krieger C-algebra C(E), and study how the distribution of loops in E aects the structure of C(E) .W e prove that C(E) is AF if and…

### Graphs, Groupoids, and Cuntz–Krieger Algebras

- Mathematics
- 1997

We associate to each locally finite directed graphGtwo locally compact groupoidsGandG(★). The unit space ofGis the space of one–sided infinite paths inG, andG(★) is the reduction ofGto the space of…

### 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…

### The C^*-algebras of infinite graphs

- Mathematics
- 1999

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