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}
}
  • A. Sims
  • Published 8 December 2003
  • Mathematics
  • Indiana University Mathematics Journal
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. 
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29 Citations
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29 Citations

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References

SHOWING 1-10 OF 10 REFERENCES

* -algebras of Finitely Aligned Higher-rank Graphs

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

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

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

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

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

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

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

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…

C*-algebras associated to higher-rank graphs

11 Chapter

Adding tails to $C^*$-correspondences

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…