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

@article{Sims2003RelativeCA, title={Relative Cuntz-Krieger algebras of finitely aligned higher-rank graphs}, author={Aidan Sims}, journal={Indiana University Mathematics Journal}, year={2003}, volume={55}, pages={849-868} }

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 for relative Cuntz- Krieger algebras.

## 29 Citations

### 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…

### A CUNTZ-KRIEGER UNIQUENESS THEOREM FOR SEMIGRAPH C*-ALGEBRAS

- Mathematics
- 2011

Higher rank semigraph algebras are introduced by mixing con- cepts of ultragraph algebras and higher rank graph algebras. This yields a kind of higher rank generalisation of ultragraph algebras. We…

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

- Mathematics
- 2011

### 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…

### 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…

### Topological higher-rank graphs and the $C^*$-algebras of topological 1-graphs

- Mathematics
- 2005

We introduce the notion of a topological higher-rank graph, a unified generalization of the higher-rank graph and the topological graph. Using groupoid techniques, we define the Toeplitz and…

### 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…

### On certain properties of Cuntz–Krieger-type algebras

- Mathematics
- 2011

The note presents a further study of the class of Cuntz--Krieger type algebras. A necessary and sufficient condition is identified that ensures that the algebra is purely infinite, the ideal…

## References

SHOWING 1-10 OF 10 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 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…

### 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).

### The ideal structure of the $C\sp *$-algebras of infinite graphs

- Mathematics
- 2001

We classify the gauge-invariant ideals in the C*-algebras of infinite directed graphs, and describe the quotients as graph algebras. We then use these results to identify the gauge-invariant…

### 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)…

### CROSSED PRODUCTS BY SEMIGROUPS OF ENDOMORPHISMS AND THE TOEPLITZ ALGEBRAS OF ORDERED GROUPS

- Mathematics
- 1994

Let r+ be the positive cone in a totally ordered abelian group F. We construct crossed products by actions of r1" as endomorphisms of C- algebras, and give criteria which ensure a given…

### 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…

### Adding tails to $C^*$-correspondences

- MathematicsDocumenta Mathematica
- 2004

We describe a method of adding tails to C*-correspondences which generalizes the process used in the study of graph C*-algebras. We show how this technique can be used to extend results for augmented…