# When is the Cuntz–Krieger algebra of a higher-rank graph approximately finite-dimensional?

@article{Evans2011WhenIT, title={When is the Cuntz–Krieger algebra of a higher-rank graph approximately finite-dimensional?}, author={D. Gwion Evans and Aidan Sims}, journal={Journal of Functional Analysis}, year={2011}, volume={263}, pages={183-215} }

## 33 Citations

### Complex Kumjian-Pask algebras

- MathematicsActa Mathematica Sinica, English Series
- 2013

Let Λ be a row-finite k-graph without sources. We investigate the relationship between the complex Kumjian-Pask algebra KPℂ(Λ) and graph algebra C*(Λ). We identify situations in which the…

### STABILITY OF C∗-ALGEBRAS ASSOCIATED TO k-GRAPHS

- Mathematics
- 2014

We give an emended proof of a result in the literature characterizing which graphs yield stable C∗-algebras. We strengthen this result by adding another necessary condition. We characterize stability…

### Some notes on complex Kumjian-Pask algebras of finitely aligned k-graphs

- Mathematics
- 2017

Suppose ۸ is a finitely aligned k-graph. In this paper we present some conditions for the k-graph ۸ in order the complex Kumjian-Pask algebra KPℂ(۸) is finite dimensional. This is an improvement for…

### Cycline subalgebras of $k$-graph C*-algebras

- Mathematics
- 2015

In this paper, we prove that the cycline subalgbra of a $k$-graph C*-algebra is maximal abelian, and show when it is a Cartan subalgebra (in the sense of Renault).

### AF-embeddability of 2-graph algebras and stable finiteness of k-graph algebras

- Mathematics
- 2015

We characterise stable finiteness of the C*-algebra of a cofinal k-graph in terms of an algebraic condition involving the coordinate matrices of the graph. This result covers all simple k-graph…

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

- MathematicsDocumenta 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…

### Continuous-trace $k$-graph $C^*$-algebras

- MathematicsRocky Mountain Journal of Mathematics
- 2018

A characterization is given for directed graphs that yield graph $C^*$-algebras with continuous trace. This is established for row-finite graphs with no sources first using a groupoid approach, and…

### Structure theory and stable rank for C*-algebras of finite higher-rank graphs

- MathematicsProceedings of the Edinburgh Mathematical Society
- 2021

Abstract We study the structure and compute the stable rank of $C^{*}$-algebras of finite higher-rank graphs. We completely determine the stable rank of the $C^{*}$-algebra when the $k$-graph either…

## References

SHOWING 1-10 OF 60 REFERENCES

### 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…

### Higher-Rank Graph C *-Algebras: An Inverse Semigroup and Groupoid Approach

- Mathematics
- 2004

AbstractWe provide inverse semigroup and groupoid models for the Toeplitz and Cuntz-Krieger algebras of finitely aligned higher-rank graphs. Using these models, we prove a
uniqueness theorem for the…

### 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…

### SIMPLICITY OF FINITELY-ALIGNED k-GRAPH C -ALGEBRAS

- Mathematics
- 2008

It is shown that no local periodicity is equivalent to the aperiodicity condition for arbitrary nitely-aligned k-graphs. This allows us to conclude that C () is simple if and only if is conal and has…

### Every AF-algebra is Morita equivalent to a graph algebra

- MathematicsBulletin of the Australian Mathematical Society
- 2004

We show how to modify any Bratteli diagram E for an AF-algebra A to obtain a Bratteli diagram KE for A whose graph algebra C*(KE) contains both A and C*(E) as full corners.

### Non-commutative spheres

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LetAθ be the irrational rotation algebra, i.e. theC*-algebra generated by two unitariesU, V satisfyingVU=e2πiθUV, with θ irrational, and consider the fixed point subalgebraBθ under the flip…

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

- Mathematics
- 2007

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…

### Higher Rank Graph C-Algebras

- Mathematics
- 2000

Building on recent work of Robertson and Steger, we associate a C{algebra to a combinatorial object which may be thought of as a higher rank graph. This C{algebra is shown to be isomorphic to that of…

### The primitive ideal space of the $C^{*}$-algebras of infinite graphs

- Mathematics
- 2002

For any countable directed graph E we describe the primitive ideal space of the corresponding generalized Cuntz-Krieger algebra C*(E).

### Cuntz-Krieger Algebras of Infinite Graphs and Matrices

- Mathematics
- 2003

We show that the Cuntz-Krieger algebras of infinite graphs and infinite {0,1}-matrices can be approximated by those of finite graphs. We then use these approximations to deduce the main uniqueness…