$C^*$-algebras associated to coverings of $k$-graphs

@article{Kumjian2006CalgebrasAT,
  title={\$C^*\$-algebras associated to coverings of \$k\$-graphs},
  author={Alex Kumjian and David Pask and Aidan Sims},
  journal={Documenta Mathematica},
  year={2006}
}
A covering of k-graphs (in the sense of Pask-Quigg-Raeburn) induces an embedding of universal C*-algebras. We show how to build a (k+1)-graph whose universal algebra encodes this embedding. More generally we show how to realise a direct limit of k-graph algebras under embeddings induced from coverings as the universal algebra of a (k+1)-graph. Our main focus is on computing the K-theory of the (k+1)-graph algebra from that of the component k-graph algebras. Examples of our construction include… 
View PDF on arXiv
18 Citations
Background Citations
4
Methods Citations
2

Figures from this paper

18 Citations

Twisted k-Graph Algebras Associated to Bratteli Diagrams

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

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…

Homology for higher-rank graphs and twisted C*-algebras

GENERALISED MORPHISMS OF k-GRAPHS: k-MORPHS

In a number of recent papers, (k + l)-graphs have been constructed from k-graphs by inserting new edges in the last l dimensions. These constructions have been motivated by C � -algebraic…

Simplicity of the C*-algebras of skew product k-graphs

We consider conditions on a $k$-graph $\Lambda$, a semigroup $S$ and a functor $\eta : \Lambda \to S$ which ensure that the $C^*$-algebra of the skew-product graph $\Lambda \times_\eta S$ is simple.…

Real rank and topological dimension of higher rank graph algebras

We study dimension theory for the $C^*$-algebras of row-finite $k$-graphs with no sources. We establish that strong aperiodicity - the higher-rank analogue of condition (K) - for a $k$-graph is…

On twisted higher-rank graph C*-algebras

We define the categorical cohomology of a k-graphand show that the first three terms in this cohomology are isomorphic to the corresponding terms in the cohomology defined in our previous paper. This…

Crossed products of k-graph C*-algebras by Zl

An action of Zl by automorphisms of a k-graph induces an action of Zl by automorphisms of the corresponding k-graph C*-algebra. We show how to construct a (k + l)-graph whose C*-algebra coincides…

On k-morphs

In a number of recent papers, (k + l)-graphs have been constructed from k-graphs by inserting new edges in the last l dimensions. These constructions have been motivated by C∗-algebraic…

COVERINGS OF SKEW-PRODUCTS AND CROSSED PRODUCTS BY COACTIONS

Abstract Consider a projective limit G of finite groups Gn. Fix a compatible family δn of coactions of the Gn on a C*-algebra A. From this data we obtain a coaction δ of G on A. We show that the…

References

SHOWING 1-10 OF 62 REFERENCES

THE C -ALGEBRAS OF ROW-FINITE GRAPHS

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 Arbitrary Graphs

To an arbitrary directed graph we associate a row-finite directed graph whose C*-algebra contains the C*-algebra of the original graph as a full corner. This allows us to generalize results for…

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…

SimpleC*-algebra generated by isometries

AbstractWe consider theC*-algebra $$\mathcal{O}_n $$ generated byn≧2 isometriesS1,...,Sn on an infinite-dimensional Hilbert space, with the property thatS1S*1+...+SnS*n=1. It turns out that…

The range of K-invariants for C*-algebras of infinite graphs

It is shown that for any pair (K 0 ,K 1 ) of countable abelian groups, with K 1 free abelian, and any element Ξ ∈ K 0 there exists a purely infinite and simple, stable C * -algebra C * (E)…

On the K-theory of 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…

CUNTZ-KRIEGER ALGEBRAS OF DIRECTED GRAPHS

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…

$C^*$-algebras of directed graphs and group actions

Given a free action of a group $G$ on a directed graph $E$ we show that the crossed product of $C^* (E)$, the universal $C^*$-algebra of $E$, by the induced action is strongly Morita equivalent to…

A class ofC*-algebras and topological Markov chains

In this paper we present a class of C*-algebras and point out its close relationship to topological Markov chains, whose theory is part of symbolic dynamics. The C*-algebra construction starts from a…

Actions of $\mathbb{Z}^k$ associated to higher rank graphs

An action of $\mathbb{Z}^k$ is associated to a higher rank graph $\Lambda$ satisfying a mild assumption. This generalizes the construction of a topological Markov shift arising from a non-negative…
...